Figure 1.1.5-5 shows the load-deflection response of the slab for the three different mesh densities using a tension stiffening value of 2 ´ 10-3. Since the coarse mesh predicts a slightly higher limit load than the medium and fine meshes do and the limit loads for the medium and fine mesh analyses are very close, the tension stiffening study is performed using the medium mesh only.
The numerical (12 ´ 12 mesh) results are compared to the experimental results in Figure 1.1.5-6 for the three different values of tension stiffening. It is clear that the less tension stiffening used, the softer the load-deflection response is. A value of tension stiffening somewhere between the highest and middle values appears to match the experimental results best. The lowest tension stiffening value causes more sudden cracking in the concrete and, as a result, the response tends to be more dynamic than that obtained with the higher tension stiffening values.
Figure 1.1.5-7 shows the numerically predicted crack pattern on the lower surface of the slab for the medium mesh.
Input files
ABAQUS/Standard input files collapseconcslab_s8r.inp
S8R elements.
collapseconcslab_s9r5.inp S9R5 elements.
collapseconcslab_postoutput.inp
*POST OUTPUT analysis.
ABAQUS/Explicit input files mcneice_1.inp
Coarse (6 ´ 6) mesh; tension stiffening = 2 ´ 10-3. mcneice_2.inp
Medium (12 ´ 12) mesh; tension stiffening = 2 ´ 10-3. mcneice_3.inp
Fine (24 ´ 24) mesh; tension stiffening = 2 ´ 10-3. mcneice_4.inp
Medium (12 ´ 12) mesh; tension stiffening = 1 ´ 10-3. mcneice_5.inp
Medium (12 ´ 12) mesh; tension stiffening = 5 ´ 10-4.
ABAQUS/Explicit results
Figure 1.1.5-5 shows the load-deflection response of the slab for the three different mesh densities using a tension stiffening value of 2 ´ 10-3. Since the coarse mesh predicts a slightly higher limit load than the medium and fine meshes do and the limit loads for the medium and fine mesh analyses are very close, the tension stiffening study is performed using the medium mesh only.
The numerical (12 ´ 12 mesh) results are compared to the experimental results in Figure 1.1.5-6 for the three different values of tension stiffening. It is clear that the less tension stiffening used, the softer the load-deflection response is. A value of tension stiffening somewhere between the highest and middle values appears to match the experimental results best. The lowest tension stiffening value causes more sudden cracking in the concrete and, as a result, the response tends to be more dynamic than that obtained with the higher tension stiffening values.
Figure 1.1.5-7 shows the numerically predicted crack pattern on the lower surface of the slab for the medium mesh.
Input files
ABAQUS/Standard input files collapseconcslab_s8r.inp
S8R elements.
collapseconcslab_s9r5.inp S9R5 elements.
collapseconcslab_postoutput.inp
*POST OUTPUT analysis.
ABAQUS/Explicit input files mcneice_1.inp
Coarse (6 ´ 6) mesh; tension stiffening = 2 ´ 10-3. mcneice_2.inp
Medium (12 ´ 12) mesh; tension stiffening = 2 ´ 10-3. mcneice_3.inp
Fine (24 ´ 24) mesh; tension stiffening = 2 ´ 10-3. mcneice_4.inp
Medium (12 ´ 12) mesh; tension stiffening = 1 ´ 10-3. mcneice_5.inp
Medium (12 ´ 12) mesh; tension stiffening = 5 ´ 10-4.
mcneice_6.inp
Medium (12 ´ 12) mesh; tension stiffening = 2 ´ 10-3; no mass scaling.
References
· Crisfield, M. A, "Variable Step-Length for Nonlinear Structural Analysis," Report 1049, Transport and Road Research Lab., Crowthorne, England, 1982.
· Gilbert, R. I. and R. F. Warner, "Tension Stiffening in Reinforced Concrete Slabs," Journal of the Structural Division, American Society of Civil Engineers, vol. 104, ST12, pp. 1885-1900, 1978.
· Hand, F. D., D. A. Pecknold, and W. C. Schnobrich, "Nonlinear Analysis of Reinforced Concrete Plates and Shells," Journal of the Structural Division, American Society of Civil Engineers, vol.
99, ST7, pp. 1491-1505, 1973.
· Hinton, E., H. H. Abdel Rahman, and O. C. Zienkiewicz, "Computational Strategies for Reinforced Concrete Slab Systems," International Association of Bridge and Structural
Engineering Colloquium on Advanced Mechanics of Reinforced Concrete, pp. 303-313, Delft, 1981.
· Lin, C. S. and A. C. Scordelis, "Nonlinear Analysis of Reinforced Concrete Shells of General Form," Journal of the Structural Division, American Society of Civil Engineers, vol. 101, pp.
523-238, 1975.
· McNeice, A. M., "Elastic-Plastic Bending of Plates and Slabs by the Finite Element Method, " Ph.
D. Thesis, London University, 1967.
Table
Table 1.1.5-1 Material properties for McNeice slab.
Concrete properties:
Properties are taken from Gilbert and Warner (1978) if available in that paper.
Properties marked with a * are not available, and are assumed values.
Young's modulus 28.6 GPa (4.15´106 lb/in2)
Poisson's ratio 0.15
Uniaxial compression values:
Yield stress 20.68 MPa (3000 lb/in2)*
Failure stress 37.92 MPa (5500 lb/in2)
Plastic strain at failure 1.5´10-3* Ratio of uniaxial tension
to compression failure stress 8.36´10-2 Ratio of biaxial to uniaxial
compression failure stress 1.16*
Cracking failure stress 459.8 lb/in2 (3.17 MPa)
Density (before mass scaling) 2.246 ´ 10-4 lb s2/in4 (2400 kg/m3)
"Tension stiffening" is assumed as a linear decrease of the stress to zero stress, at a strain of 5´10-4, at a strain of 10´10-4, or at a strain of 20´10-4.
mcneice_6.inp
Medium (12 ´ 12) mesh; tension stiffening = 2 ´ 10-3; no mass scaling.
References
· Crisfield, M. A, "Variable Step-Length for Nonlinear Structural Analysis," Report 1049, Transport and Road Research Lab., Crowthorne, England, 1982.
· Gilbert, R. I. and R. F. Warner, "Tension Stiffening in Reinforced Concrete Slabs," Journal of the Structural Division, American Society of Civil Engineers, vol. 104, ST12, pp. 1885-1900, 1978.
· Hand, F. D., D. A. Pecknold, and W. C. Schnobrich, "Nonlinear Analysis of Reinforced Concrete Plates and Shells," Journal of the Structural Division, American Society of Civil Engineers, vol.
99, ST7, pp. 1491-1505, 1973.
· Hinton, E., H. H. Abdel Rahman, and O. C. Zienkiewicz, "Computational Strategies for Reinforced Concrete Slab Systems," International Association of Bridge and Structural
Engineering Colloquium on Advanced Mechanics of Reinforced Concrete, pp. 303-313, Delft, 1981.
· Lin, C. S. and A. C. Scordelis, "Nonlinear Analysis of Reinforced Concrete Shells of General Form," Journal of the Structural Division, American Society of Civil Engineers, vol. 101, pp.
523-238, 1975.
· McNeice, A. M., "Elastic-Plastic Bending of Plates and Slabs by the Finite Element Method, " Ph.
D. Thesis, London University, 1967.
Table
Table 1.1.5-1 Material properties for McNeice slab.
Concrete properties:
Properties are taken from Gilbert and Warner (1978) if available in that paper.
Properties marked with a * are not available, and are assumed values.
Young's modulus 28.6 GPa (4.15´106 lb/in2)
Poisson's ratio 0.15
Uniaxial compression values:
Yield stress 20.68 MPa (3000 lb/in2)*
Failure stress 37.92 MPa (5500 lb/in2)
Plastic strain at failure 1.5´10-3* Ratio of uniaxial tension
to compression failure stress 8.36´10-2 Ratio of biaxial to uniaxial
compression failure stress 1.16*
Cracking failure stress 459.8 lb/in2 (3.17 MPa)
Density (before mass scaling) 2.246 ´ 10-4 lb s2/in4 (2400 kg/m3)
"Tension stiffening" is assumed as a linear decrease of the stress to zero stress, at a strain of 5´10-4, at a strain of 10´10-4, or at a strain of 20´10-4.
Steel (rebar) properties:
Young's modulus 200 GPa (29´106 lb/in2)
Yield stress 345 MPa (50´103 lb/in2)
Density (before mass scaling) 7.3 ´ 10-4 lb s2/in4 (7800 kg/m3)
Figures
Figure 1.1.5-1 McNeice slab.
Figure 1.1.5-2 Load-deflection response of McNeice slab, ABAQUS/Standard.
Steel (rebar) properties:
Young's modulus 200 GPa (29´106 lb/in2)
Yield stress 345 MPa (50´103 lb/in2)
Density (before mass scaling) 7.3 ´ 10-4 lb s2/in4 (7800 kg/m3)
Figures
Figure 1.1.5-1 McNeice slab.
Figure 1.1.5-2 Load-deflection response of McNeice slab, ABAQUS/Standard.
Figure 1.1.5-3 Crack pattern on lower surface of slab, ABAQUS/Standard.
Figure 1.1.5-4 Load-deflection response of McNeice slab, ABAQUS/Explicit; influence of mass scaling.
Figure 1.1.5-3 Crack pattern on lower surface of slab, ABAQUS/Standard.
Figure 1.1.5-4 Load-deflection response of McNeice slab, ABAQUS/Explicit; influence of mass scaling.
Figure 1.1.5-5 Load-deflection response of McNeice slab, ABAQUS/Explicit; influence of mesh refinement.
Figure 1.1.5-6 Load-deflection response of McNeice slab, ABAQUS/Explicit; influence of tension stiffening.
Figure 1.1.5-5 Load-deflection response of McNeice slab, ABAQUS/Explicit; influence of mesh refinement.
Figure 1.1.5-6 Load-deflection response of McNeice slab, ABAQUS/Explicit; influence of tension stiffening.
Figure 1.1.5-7 Crack pattern on lower surface of slab, ABAQUS/Explicit.
Sample listings
Figure 1.1.5-7 Crack pattern on lower surface of slab, ABAQUS/Explicit.
Sample listings
Listing 1.1.5-1
*HEADING
CORNER SUPPORTED TWO-WAY SLAB TESTED BY MCNEICE
CORNER SUPPORTED TWO-WAY SLAB TESTED BY MCNEICE
GEOMETRY=ISOPARAMETRIC,NAME=XX
Listing 1.1.5-2
*HEADING
CORNER SUPPORTED TWO-WAY SLAB TESTED BY MCNEICE
CORNER SUPPORTED TWO-WAY SLAB TESTED BY MCNEICE
4.15E6,.15
*HISTORY OUTPUT, TIME= 5.00E-4
*ENERGY HISTORY
ALLKE, ALLSE, ALLIE, ALLWK, ALLAE, ETOTAL
*NODE HISTORY,NSET=ONE
*HISTORY OUTPUT, TIME= 5.00E-4
*ENERGY HISTORY
ALLKE, ALLSE, ALLIE, ALLWK, ALLAE, ETOTAL
*NODE HISTORY,NSET=ONE
1.1.6 Jointed rock slope stability
Product: ABAQUS/Standard
This example illustrates the use of the jointed material model in the context of geotechnical applications. We examine the stability of the excavation of part of a jointed rock mass, leaving a sloped embankment. This problem is chosen mainly as a verification case because it has been studied previously by Barton (1971) and Hoek (1970), who used limit equilibrium methods, and by
Zienkiewicz and Pande (1977), who used a finite element model.
Geometry and model
The plane strain model analyzed is shown in Figure 1.1.6-1together with the excavation geometry and material properties. The rock mass contains two sets of planes of weakness: one vertical set of joints and one set of inclined joints. We begin from a nonzero state of stress. In this problem this consists of a vertical stress that increases linearly with depth to equilibrate the weight of the rock and horizontal stresses caused by tectonic effects: such stress is quite commonly encountered in geotechnical
engineering. The active "loading" consists of removal of material to represent the excavation. It is clear that, with a different initial stress state, the response of the system would be different. This illustrates the need of nonlinear analysis in geotechnical applications--the response of a system to external
"loading" depends on the state of the system when that loading sequence begins (and, by extension, to the sequence of loading). We can no longer think of superposing load cases, as is done in a linear analysis.
Practical geotechnical excavations involve a sequence of steps, in each of which some part of the material mass is removed. Liners or retaining walls can be inserted during this process. Thus,
geotechnical problems require generality in creating and using a finite element model: the model itself, and not just its response, changes with time--parts of the original model disappear, while other
components that were not originally present are added. This example is somewhat academic, in that we do not encounter this level of complexity. Instead, following the previous authors' use of the example, we assume that the entire excavation occurs simultaneously.