Dynamic Increase Factor (DIF) due to strain rate effect was examined with documented experimental work done by Razaqpur, et al. In the experiment work, two 1000 × 1000 × 70 mm reinforced concrete slabs were constructed. The slabs were subjected to blast loads generated by the detonation of either 22.4 kg or 33.4 kg of ANFO located at a 3.0 m standoff. Blast wave characteristics, including incident and reflected pressures and reflected impulses were measured. The slabs were modeled by explicit analysis with or without strain rate effect to study their behavior under blast load to compare their predicted and observed behavior. The predicted post-blast damage and mode of failure for each model is compared with the observed damage of experimental work. It was concluded that when the dynamic increase factor added to concrete and reinforcement materials due to strain rate effect, the behavior of model under blast load become closer to experimental work.
Since testing of structures under the effect of real explosives requires complex instrumentation and a safe test range, it is not always feasible to carry out a large number of such tests. Therefore, to gain deeper insight into the detailed behavior and performance of structures under blast loads, one must resort to analytical and advanced numerical techniques. However, the results of the analysis must be confirmed by some amount of testing to en- sure the validity of the assumptions and procedures used in the analysis.
The objective of this paper is to compare the observed behavior of reinforced concrete panels, subjected to nominally similar blast loads, with their predicted behavior using explicit analysis computer program Ls-Dyna for the following material models:
1) Static stress-strain curve for concrete and steel material;
2) Stress-strain curve for concrete and steel material in conjunction with strain rate effect.
Explicit FEM analysis is used to apply incremental procedure for load (or displacement). At the end of each increment, the stiffness matrix based on geometry changes and material changes (if applicable) is updated. Then a new stiffness matrix is constructed, and the next increment of load (or displacement) is applied to the system. This method does not enforce equilibrium of the internal structure forces with the externally applied loads. Therefore, the hope is that if the increments are small enough, the results will be accurate.
When the loading rate is high, the mechanical response of a material is generally different from that at a low load- ing rate. Such rate dependence is observed for nearly all the brittle materials. Concrete exhibits also an enigmatic phenomenon of increased resistance when it is loaded at a very high rate as explained below.
Seven 1000 × 1000 × 70 mm reinforced concrete slabs (
The slabs designated CS2 to CS4 were as-built while other panels were retrofitted on each face with two sheets of GFRP [
To commence the test, the tripod holding the charge was centered above the center of the panel and the charge was hung with a wire. The distance from the center of the charge to the center of the test panel was 3.0 m for all panels. The explosive used was ANFO, comprising 5.7% fuel oil and 94.3% ammonium nitrate, shaped into an approximately spherical form. The explosive energy of ANFO is 3717 kJ/kg, which is 82% of the energy of one kilogram of TNT [
It should be noted that specimen CS4 was subjected to the pressure produced by a 22.4 kg charge while the remaining specimens were exposed to the load caused by the detonation of a 33.4 kg charge. In this study, the results of specimens CS2 and CS3 are considered in the comparison with their counterpart of the predicted ex- plicit analysis, as the post-blast observed damage in the un-retrofitted test panels was available for CS2 and CS3.
In both models, the slab support was assumed to be hinged. Blast loading was calculated using the empirical blast loading functions implemented in the CONWEP code based on TM5-1300 technical manual [
In this model, Kinematic Hardening Cap Model was used for concrete material. The implementation of an ex- tended two invariant cap model, suggested by Stojko is based on the formulations of Simo, et al. and Sandler & Rubin [
One of the major advantages of the cap model over other classical pressure-dependent plasticity models is the ability to control the amount of plastic volumetric strain (dilatency) produced under shear loading. Dilatency is produced under shear loading as a result of the yield surface having a positive slope in
dilatency continues as long as shear loads are applied, and in many cases produces far more dilatency than is experimentally observed in material tests. In the cap model, when the failure surface is active, dilatency is pro- duced just as with the Drucker-Prager and Mohr-Columb models.
However, the hardening law permits the cap surface to contract until the cap intersects the failure envelope at the stress point, and the cap remains at that point. The local normal to the yield surface is now vertical, and therefore the normality rule assures that no further plastic volumetric strain (dilatency) is created. In this model, the concrete material fails under principal strain for concrete material. Model 1 is shown in
Reinforcement representation in this model was discrete reinforcement in solid elements. Elastic plastic with kinematic hardening material is used for reinforcement bars. The rebar are capable of tension and compression, but not shear (see
Experimentally, it was observed on the bottom surface of most panels an array of 500 mm long cracks formed a square shape centered on the panel center and propagated diagonally towards the corners of the panel. Also, cracks inside the square were noted. These cracks are similar to the yield line pattern for a statically applied cen- tral patch load. On the bottom surface, additional minor cracks, which typically followed the reinforcement layout, were also observed in experiment work.
Typically, all damaged panels had full depth inverted 45 shear cracks near their supports and on all four sides. These cracks were rather wide and in some cases greater than 4 mm [
Analytically, the finite element model fails prematurely. The model show severe damage and most elements failed and deleted. But it is possible to track damage propagation within blast period. On bottom (tension) side, cracks start propagating at support edges (see
Also, on the bottom surface, cracks followed the reinforcement layout, were also observed, which means that the bond between concrete and steel are demolished.
On the compression face, cracks propagate latterly. Stresses are similar to yield line pattern for a statically applied central patch load (see
Finally central cracks were met with diagonal cracks as seen in
Cracks propagation is similar to experimental work as seen in
Maximum displacement was at the center of plate but it cannot be estimated here because of the total loss for panel stiffness. Complete demolition for concrete plate may refer to the strain rate effect. When the loading rate is high, the mechanical response of a material is generally different from that at a low loading rate.
In this model, Piecewise Linear Isotropic Plasticity material is used. This material includes strain rate effects [
Elastic plastic with kinematic hardening material is used for reinforcement bars (see
Experimentally, it was observed on the bottom surface of most panels an array of 500 mm long cracks formed
a square shape centered on the panel center and propagated diagonally towards the corners of the panel.
Also, cracks inside the square were noted. These cracks are similar to the yield line pattern for a statically ap- plied central patch load.
On the bottom surface, additional minor cracks, which typically followed the reinforcement layout, were also observed in experimental model. Analytically, in the bottom “tension” face, cracks are propagates in center and extend to the edges in the same time (see
In the top “compression” face, diagonal cracks are propagated (see
In Ls-Dyna model, the size of square shape array is 550 mm long and the damage in yield line region extend to the 50% of the plate thickness. Addition damages are also noted at support boundary (see
The maximum experimental central deflection for the two identical plates was 13.12 mm for panel CS2 and 9.53 mm for panel CS3, with an average deflection of 11.33 mm. Maximum central deflection in Model 2 was 11.66 mm as seen in
Damage pattern in Ls-Dyna model 2 is more close to experimental work as seen in
The comparison between the field test results for the slabs subjected to a detonation of 33.4 kg of (ANFO) ma-
terial generally compared well with the results of the explicit analysis using Ls-Dyna. it is possible to study the model after total failure where the model become unstable, i.e. explicit solver provide better presentation for blast load than implicit solver.
It was also concluded that strain rate effect is vital to get good presentation for blast load. Comparison proves that using Dynamic Increase Factor (DIF) for concrete and steel material provides much better presentation. So, it is important to account for dynamic increase factor for concrete and steel material for high strain rate loads such as blast and impact.