Loads and displacement - initial vertical loading

The initial stiffness of the frame shows a clear elastic behaviour. At the appearance of cracking, the stiffness gradually decreases during increased loading until more than half of the characteristic horizontal load is applied. From here, the cracking seems to have stabilized and the stiffness becomes more linear again. The stiffness undergoes one last great reduction at (or right before) concrete failure of the top right corner. All this is visible in the load-displacement curves given in Figs. 3.23 to 3.28. Note that the vertical displacement increases during load sequence 2. Although the vertical load-displacement plot of LH7 is almost identical to LH4, it is included to show the PSFm curve does not reach design load levels, as it diverges at 1026 kN.

Because the vertical load of LH4 and LH7 does not increase above design loading, the load-displacement plots have been restricted to the load sequences where vertical load is increasing. The resulting changes in vertical displacement is therefore not visible in the Figs. 3.25 and 3.27. In LH4, the vertical displacement remains constant or experience a slight increase for all analyses as horizontal load is increased above characteristic loading. During load sequence 4, PSFm diverges before design values are reached, and vertical displacement for GRFm does not change before local failure, but increases after failure. For the mean ECOV analysis, vertical displacement is reduced when horizontal load exceeds design level which continues until global failure. The total reduction amounts to roughly 15% compared to the displacement at design load level. Meanwhile, characteristic ECOV displacement does not change until global failure.

Above design load level, LH7 experienced similar behaviour as LH4, and mean ECOV analy-sis experienced a displacement reduction of roughly 14%. In addition, PSFm analyanaly-sis did not reach design levels, characteristic ECOV analysis retained a constant vertical displacement, and GRFm analysis failed locally in the right corner before increasing the vertical displace-ment.

For the load histories with initial vertical loading, all safety format methods except the GRFm diverged when the interior of the top right corner failed in concrete compression.

An illustration of such a crushing can be seen in Fig. 3.29. The divergence of the PSFm and characteristic ECOV analysis for LH1 stands in contrast to the verification analyses, where the same analyses did not diverge at first concrete failure. The more brittle behaviour does not affect the PSFm capacity to a significant degree compared to the verification analyses, however, the stiffness is greatly increased due to a large increase of the stiffness modulus of steel and concrete. Meanwhile, the characteristic ECOV capacity is slightly reduced compared to the verification analyses as a result of the tensile strength of concrete and ultimate strain of reinforcement being reduced.

As the GRFm analyses do not diverge at the concrete failure of the right corner, the capacity is increased due to a redistribution of forces. At failure, the right hand corner becomes a plastic hinge and looses some of its strength. Consequently, the load effects must be carried by other parts of the frame. The vertical loading is being carried by the beam which now more represents a simply supported beam which increases the compressive stresses in the mid span. Likewise, the horizontal load is carried to a greater extent by the columns which acts more like counter lever beams fixed at the supports. As the loading continues, the stress increase at the supports is significant. This redistribution increases the global capacity with roughly 20%. Illustrations are given in Fig. 3.30 which show the principal compressive

stresses for the GRFm analysis of LH1 right before compressive failure, and Fig. 3.31 which show the principal compressive stresses from the same analysis right before global failure.

0 5 10 15 20 25 30 35 40 45

Figure 3.23: Vertical force, F_V, vs. vertical displacement,v, in mid span of the frame for all NLFEA (see legend) of load history 1. Load sequence 2 is left out for clarity which causes the horizontal shift at 800 kN.

0 10 20 30 40 50 60 70 80 90 100

Figure 3.24:Horizontal force,FH, vs. horizontal displacement,u, of the top right corner of the frame for all NLFEA (see legend) of load history 1. Load sequence 1 is left out for clarity which causes the small initial shift from the origin.

0 5 10 15 20 25

Figure 3.25: Vertical force, FV, vs. vertical displacement,v, in mid span of the frame for all NLFEA (see legend) of load history 4. Load sequence 2 and 4 is left out for clarity which causes the horizontal shift at 800 kN and the cut off at 1080 kN respectively.

0 20 40 60 80 100 120

Figure 3.26:Horizontal force,FH, vs. horizontal displacement,u, of the top right corner of the frame for all NLFEA (see legend) of load history 4. Load sequence 1 and 3 is left out for clarity which causes the small initial shift from the origin and the horizontal shift at 675 kN respectively.

0 5 10 15 20 25 30 v [mm]

0 200 400 600 800 1000 1200

F V [kN]

LH7: Vertical force vs. displacement

PSFm GRFm ECOV mean ECOV char

Figure 3.27: Vertical force, FV, vs. vertical displacement,v, in mid span of the frame for all NLFEA (see legend) of load history 7. Load sequence 2 and 4 is left out for clarity which causes the horizontal shift at 800 kN and the cut off at 1080 kN respectively.

0 20 40 60 80 100 120

u [mm]

0 200 400 600 800 1000 1200 1400

F H [kN]

LH7: Horizontal force vs. displacement

PSFm GRFm ECOV mean ECOV char

Figure 3.28:Horizontal force,FH, vs. horizontal displacement,u, of the top right corner of the frame for all NLFEA (see legend) of load history 7. Load sequence 1 is left out for clarity which causes the small initial shift from the origin.

Figure 3.29: Illustration of concrete compressive failure in the column at the right frame corner. From GRFm analysis for load history 1. The figure show the principal compressive strainsε2(E2 in figure) at a time after compressive failure of the top right frame corner.

Figure 3.30: Illustration of principal concrete compressive stresses (S2, measured in MPa) in the frame right before concrete failure of the top right corner. From GRFm analysis for load history 1.

Figure 3.31: Illustration of principal concrete compressive stresses (S2, measured in MPa) in the frame right before global failure. From GRFm analysis for load history 1.