The seismic analysis and performance of steel frame with additional low-yield-point steel dampers

3.1.1. Period

We can observe that the steel frame without LYP steel dampers has a higher period compared to the steel frame with LYP steel dampers. With the introduction of the LYP steel damper, there is a reduction in the period because the damper increases the stiffness of the steel frame. This shows that the addition of LYP steel dampers reduces the seismic response on the steel frame. Fig. 7 shows the periods of two steel frames.

Fig. 7Period of the two steel frames

Fig. 8Frequency of the two steel frames

3.1.2. Frequency

The frequency is inversely proportional to the period, so lower periods would generally have higher frequencies and vice versa. So the steel frame with LYP steel dampers will have a larger frequency compared to the steel frame without LYP steel dampers. The chart comparing the two different steel frames is shown in Fig. 8.

4.1.1. Amplitude adjustment for earthquake waves

As stated in the CCFSDB maximum acceleration should be adopted for different seismic waves in the time history analysis. The maximum acceleration of the three types of earthquake level in China (i.e. frequent, moderate and rare) under fortification intensity of 8 is shown in Table 6. Eq. (2) is used to change the earthquake record maximum acceleration, time-history graph under their respective seismic levels:

where $a^{\text{'}}\left(t\right)$ is the adjusted seismic acceleration graph, $A_{max}$ is the maximum value of acceleration in the time-history graph under different seismic intensity as stated in the code for the sdob, $a\left(t\right)$ defines the original seismic record. Time-history graph of the three different seismic intensity levels as stated in the code for the seismic design of buildings.

Note: values in the bracket are used that the design basic acceleration of ground motion is 0.15 g and 0.30 g respectively. The representative seismic waves and corresponding frequency spectra of PGA = 300 gal as an example of these three ground records are shown in Fig. 11 and Fig. 12. The dominant frequencies of adopted El Centro and Taft earthquake waves are 1.82 Hz and 1.72 Hz, respectively. Under the seismic mechanism analysis, we should try and find the stress distribution, top story displacement, inter-story drift, and base shear, frame’s forces and the damper’s stress. First steel frame without steel dampers is analyzed and the results are studied. The structure was modelled using the SAP2000 software.

Fig. 11Loading earthquake waves

a) El Centro wave

b) Taft wave

c) Artificial wave

Fig. 12Corresponding frequency spectra

a) El Centro wave

b) Taft wave

c) Artificial wave

4.2.1. Displacement of the top story

Top story displacement is obtained for the respective earthquake waves. Time displacement graph is shown in Fig. 13 and 14 of the respective earthquake waves for both steel frames. The algorithm for computing the time history analysis is the HHT method. The HHT method or alpha method is used in the structural dynamics for the numerical integration of a linear set of second ODE. The SAP2000 software would use the HHT method in performing the time history analysis. The maximum displacement in the steel structure without LYP steel dampers top story for the different earthquake waves under the three different earthquake levels are shown in Fig. 13. Table 7 shows the maximum displacement values of the structure without LYP steel dampers in the horizontal direction or $X$ direction.

Fig. 13Displacement of earthquake waves under the three earthquake levels for the steel frame without LYP steel dampers

a) El Centro frequent

b) El Centro moderate

c) El Centro rare

d) Taft frequent

e) Taft moderate

f) Taft rare

g) Artificial frequent

h) Artificial moderate

i) Artificial rare

Fig. 14Displacement of earthquake waves under the three earthquake levels for the steel frame with LYP steel dampers

a) El Centro frequent

b) El Centro moderate

c) El Centro rare

d) Taft frequent

e) Taft moderate

f) Taft rare

g) Artificial frequent

h) Artificial moderate

i) Artificial rare

The steel frame with LYP steel dampers was also analyzed, the maximum displacement in the steel structure with LYP steel dampers top story is obtained and the displacement of the respective earthquake waves under the three earthquake levels are shown below in Fig. 14. Table 8 shows the maximum displacement values of the steel frame with LYP steel dampers in the horizontal direction. We can observe that with the addition of LYP steel dampers, top story’s displacement reduction is observed in frequent, moderate and rare earthquake. This conforms to the fact that the addition of and Artificial waves under the three earthquake levels respectively. The Artificial earthquake wave produces less seismic influence on the steel frame when compared to the other earthquake waves. The maximum displacements of the top story of the two steel frames under the three earthquake ground motions are shown in Fig. 15.

Fig. 15Maximum displacement of earthquake waves under the three earthquake levels

a) Top story frequent earthquake level

b) Top story moderate earthquake level

c) Top story rare earthquake level

The LYP steel dampers reduce the seismic response of the steel frame. Frequent type earthquake produces lower displacements on the structures as compared to the rare earthquake, this shows that the rare earthquake produces more seismic force on the structure. The maximum displacement in the top story is reduced by 62.8 %, 59.3 % and 48.4 % in the El Centro, Taft and Artificial waves under the three earthquake levels respectively.

4.2.2. Inter-story drift angle

Inter-story drift is an important indicator of structural behavior in performance-based seismic analysis; inter-story drift is one of the particularly useful engineering response quantity and indicator of structural behavior especially for high-rise buildings [29]. Story drift angle is one of the main indexes to test the seismic performance of building structures [30]. The steel frame examined in this study is a two-story steel frame-structure. The frame that is being studied here is not a high-rise structure but it is necessary to study the inter-story drift to get a better understanding of the structural behavior of the structure show in Fig. 16.

Fig. 16Inter-story drift on the structure

Inter-story drift is an important indicator of structural behavior in performance-based seismic analysis; inter-story drift is one of the particularly useful engineering response quantity and indicator of structural behavior especially for high-rise buildings [29]. Story drift angle is one of the main indexes to test the seismic performance of building structures [30]. The steel frame examined in this thesis is a two-story steel frame-structure. The frame that is being studied here is not a high-rise structure but it is necessary to study the inter-story drift to get a better understanding of the structural behavior of the structure.

First floor’s Inter-story drift angle is defined as:

Second floor’s Inter-story drift angle is defined as:

where ${\theta }_i$ – inter-story drift angle; ${∆}_i$ – displacement, $H_i$ – floor height.

Table 9 shows the inter-story drift angle for the steel frame without LYP steel damper and their respective earthquake waves.

The floor height of each floor is 3200 mm.

1) For El Centro earthquake wave.

2) For Taft earthquake wave.

3) For the Artificial earthquake wave.

4.2.3. Steel frame with LYP steel damper

We obtained the inter-story drift from Eqs. (3-4) for the steel frame with LYP steel dampers.

The floor height of each floor is 3200 mm.

1) For El Centro earthquake wave.

2) For Taft earthquake wave.

3) For the Artificial earthquake wave.

The maximum limit given in the China seismic codes is 1/250 = 0.004 under frequent earthquake and 1/50 = 0.02 under rare earthquake [41]. There is inter-story’s reduction in drift angle with LYP steel damper in the steel frame. The inter-story’s reduction drift in the second floor is studied. Inter-story drift’s reduction in the frequent earthquake type 81 %, 77 % and 61 % in the El Centro, Taft and Artificial earthquake waves is observed by introducing the LYP steel dampers in the steel frame. we noticed a reduction in inter-story drift in the moderate earthquake level by 86 %, 56 % and 63 % in the El Centro, Taft and Artificial earthquake waves respectively by introducing the LYP steel dampers in the steel frame, we noticed inter-story’s reduction drift in the rare earthquake type by 74 %, 67 % and 62 % in the El Centro, Taft and Artificial earthquake waves by introducing the LYP steel dampers in the steel frame. This shows that with the introduction of LYP steel dampers, we can reduce the inter-story drift in the steel frame if the steel frame was a high-rise structure, we would be able to notice that the inter-story drift of the steel frame without LYP steel dampers would most likely have an inter-story drift greater than the maximum limit given in the China seismic codes, but since the steel frame being studied here is a two-story frame, we were not able to see the inter-story drift angle of the steel frame exceed the maximum limit. The distribution of the inter-story drift angle of the two steel frames under the three earthquake level is shown in Fig. 17.

Fig. 17Inter-story drift angle under the three earthquake levels

a) First floor frequent earthquake level

b) Second floor frequent earthquake level

c) First floor moderate earthquake level

d) Second floor moderate earthquake level

e) First floor rare earthquake level

f) Second floor rare earthquake level

4.2.4. Stress of dampers

This section shows the stress in the LYP steel dampers, Von Mises stress is one of the dampers’ type stress. In order to determine whether given material is yield or fracture Von Mises stress value is used for its evaluation. Von Mises state of stress on a 3D object showing the principal in Fig. 18. In SAP2000 the von Mises stress provides a measure if the stress is shear, or distortional in the material. Eqs. (5, 6) shows the on Mises stress in 3D. The derivation of this form of the von Mises stress is based on the principal axes and arrives at the final result that is, of course, expressed in terms of the principal stresses ${\sigma }_1$, ${\sigma }_2$ and ${\sigma }_3$ [32] in Eq. (5):

The general form of the von Mises stress:

Fig. 18Von Mises state of stress on a 3D object showing the principal

For the El Centro frequent, moderate and rare earthquake level the maximum stress in the LYP steel damper are 128.99 MPa, 209.69 MPa and 209.73 MPa respectively, we can observe that the stress are almost similar in all the LYP steel. The Taft frequent, moderate and rare earthquake level the values of the maximum stress in the LYP steel damper are 129.97 MPa, 209.51 MPa and 209.77 MPa. The maximum stress in the damper is 154.29 MPa for the artificial earthquake frequent level, the LYP steel damper on the right-hand side experience the greatest stress while stress in the middle area of the damper is greatest and the stress reduces as the dampers connects to both the top and bottom plates. This shows that the LYP steel dampers are good seismic energy dissipation agent. For the Artificial earthquake moderate level, all the LYP steel dampers are in the elastic-plastic stage and the maximum stress on the dampers is 209.62 MPa.

For the Artificial earthquake rare level, the maximum stress is 209.81 MPa, this shows that the energy dissipation of the LYP steel damper has a good effect on the steel frame. We can observe that the stress of the LYP steel dampers increases as the earthquake level increases, this shows that the LYP steel dampers are good energy dissipation agents.

Fig. 19Damper stress of the earthquake waves under the three earthquake levels for the steel frame with LYP steel dampers (Source: Author’s own calculations Trough Sap2000)

a) El Centro frequent

b) El Centro moderate

c) El Centro rare

d) Taft frequent

e) Taft moderate

f) Taft rare

g) Artificial frequent

h) artificial moderate

i) Artificial rare

4.3.1. Axial force

An axial force is the tension or compression which acts on a frame or a member of a structure. Tensile forces are positive while compression forces are negative. We compared the axial forces in the members of the two different steel frames. The axial force acting on the beams of the steel frame without LYP steel dampers has greater force as compared to the steel frame without steel dampers. It conforms that LYP steel dampers reduces the internal forces acting on the members of the structure. The comparison of the axial force of both beam and column member of the two steel frames is shown in Fig. 24.

Fig. 22Combination 2 (1.2(DL + 0.5LL) + 1.3Ek) bending moment diagram of the steel structure without LYP dampers (Newton-mm)

Fig. 23Combination 2 (1.2(DL + 0.5LL) + 1.3Ek) bending moment diagram of the steel structure with LYP dampers (Newton-mm)

Fig. 24Maximum axial force of members

a) Column section AE

b) Beam section BH

4.3.2. Shear force

Shear force acts on a member or frame in perpendicular or horizontal direction. For its design, the capacity to resist shear forces is important than to resist axial forces. In Fig. 25, we observed a reduction in the shear force in beam BH. It shows that the addition of the dampers increases the member’s resistance of the steel frame.

Fig. 25Maximum shear force of members

a) Shear force of column section AE

b) Beam section BH

4.3.3. Bending moment

The reaction which is produced in a structural member when an external force is applied which cause some bending of the member, this phenomenon is known as bending moment. We can notice a reduction in the internal forces which acts on the steel frame with LYP steel dampers when it is compared to the steel frame without LYP steel dampers. This shows that the addition of LYP steel dampers causes a reduction in the bending. Fig. 26 describes the bending moments in the steel frame with LYP steel dampers are closer to the zero mark as compared to the steel frame without LYP steel dampers. It represents the introduction of dampers helps to reduce the bending moment in the members of the steel frame.

Fig. 26Maximum bending moment of members

a) Bending moment of column section AE

b) Bending moment of beam section BH

4.3.4. Stress of damper

We would use the three different load combinations to get the stress in the dampers of the steel frame with LYP steel dampers. This is to make sure we check for the design requirements of the steel frame with LYP steel dampers. From Fig. 27, the maximum stress in the damper for the combination 1 (1.2DL + 1.4LL) is 53.80 MPa, we notice that the two outer LYP steel dampers experience more stress compared to the two inner LYP steel dampers, the two outer LYP steel dampers are within the range of 17-35 MPa which is within the elastic-plastic stage and has begun to consume energy, the LYP steel dampers are used as a defense mechanism to prevent the steel frame from destruction. The maximum stress in Fig. 28 for combination 2 (1.2(DL + 0.5LL) + 1.3Ek) for the LYP steel dampers is 221.99 MPa.

Fig. 27Stress of dampers for Combination 1 (1.2DL + 1.4LL) (MPa)

Fig. 28Stress of dampers for Combination 2 {1.2(DL + 0.5LL) + 1.3Ek}

4.4.1. Capacity spectrum method (ATC-40)

The Capacity Spectrum Method (CSM) was developed by S. A. Freeman for frame buildings, its concept have been introduced in several United States guidelines for seismic evaluations such as ATC-40 and the NEHRP guidelines for the seismic rehabilitation of buildings [36] show in Fig. 31. In this method the maximum inelastic deformation of a nonlinear SDOF system can be approximated from the maximum deformation of a linear elastic SDOF system with an equivalent period and damping [37]:

Fig. 31Schematic representation of Capacity Spectrum Method (ATC-40)

4.5.1. The base shear displacement curve of the two structures

Using the base shear displacement curve, we are able to get the structural stiffness of both steel frames show in Fig. 32. This will help us understand how the LYP steel dampers affect the structural stiffness of the steel frame ATC-40.

Fig. 32Structural stiffness ATC-40

The structural stiffness is the slope of the pushover curve at the elastic stage. Structural stiffness gotten from the pushover curve formula is, we can get the structural stiffness of both steel frames:

where $V_{b_u}$ – base shear from point B to IO in the pushover curve, $u_u$ – displacement from B to IO in the pushover curve.

From [11], we can get the structural stiffness of both steel frames.

For uniform load for steel frame without LYP steel damper:

For triangle load pushover curve for steel frame without LYP steel damper:

For uniform load for steel frame with LYP steel damper:

For triangle load pushover curve for steel frame with LYP steel damper:

Fig. 33The based curve of the two steel frames (Source: Author’s own Figure Trough Sap2000)

a) Uniform load

b) Triangle load

The comparison on the two loads analyzed in the pushover analysis shows that the steel frame without LYP steel dampers is safe since it did not collapse while for the uniform pushover curve, we noticed that the frame entered the yield zone very early; a further check on the structure shows that the braces yielded; this conforms to the fact that the braces and the LYP steel dampers are the first line of defense against structural failure. The structural stiffness of the steel frame with LYP steel dampers is greater than the steel frame without LYP steel dampers as can be observed from the higher curve as seen from the pushover curves for the uniform load and triangle load .in the uniform pushover curve of the steel frame with LYP steel damper, we noticed the frame yielded early. The stiffness has an increase of 44 % for both the uniform and triangle load pushover curve with the addition of LYP steel dampers to the steel frame. The higher structural stiffness shows that the addition of LYP steel dampers to the steel frame helps the steel frame increase its seismic resistance.

4.5.2. Inter-story drift angle and comparison with the inter-story drift angle results of the time history analysis

Inter-story drift is one of the most important aspect in the seismic design of a structure, we would present the inter-story drift angle from the ATC-40 capacity spectrum and then compare it with the inter-story drift angle gotten from the time history analysis. The performance of a structure based on the drift limit is presented in Table 16.

The inter-story drift angle values obtained from the ATC-40 capacity spectrum is shown in Table 17 and Table18 for their respective lateral loads for the steel frame without LYP steel dampers.

The inter-story drift angle values obtained from the ATC-40 capacity spectrum is shown in Table 19 and 20 for their respective lateral loads for the steel frame with LYP steel dampers.

We would ignore the uniform ATC-40 capacity spectrum inter-story drift angle since the frame yielded and it would not give us the required results under the different earthquake types, we would focus solely on the triangle ATC-40 capacity spectrum. According to the seismic codes for the design of buildings is 1/250 = 0.004 under frequent earthquake and 1/50 = 0.02 under rare earthquake. From Fig. 34 and Fig. 35 for both ATC-40 capacity spectrum and the time-history analysis, we can see that the maximum inter-story drift angle value is 0.0016 for the frequent earthquake type and 0.0083 for the rare earthquake type. Based on the performance level of structures, we can notice that for frequent, moderate and rare earthquake type, the building was not damaged, slightly damaged and moderately damaged respectively; this shows that the building satisfies the specifications as stated in the code for the seismic design of buildings.

Fig. 34Inter-story drift comparing the inter-story drift in time-history analysis and that of ATC 40 capacity spectrum for the steel frame without LYP steel dampers

a) Frequent earthquake motion

b) Moderate earthquake motion

c) Rare earthquake motion

Fig. 35Inter-story drift comparing the inter-story drift in time-history analysis and that of ATC 40 capacity spectrum for the steel frame with LYP steel dampers

a) Frequent earthquake motion

b) Moderate earthquake motion

c) Rare earthquake motion

4.5.3. Stress of damper and comparison with the base stress of damper results of the time history analysis

We would get the von Mises equivalent stress which is the force per unit area in the LYP steel dampers in the uniform and triangle load pushover analysis. In the uniform load ATC-40 capacity spectrum, for frequent, moderate and rare earthquake level, the maximum yield stress reaches up to 130.4 MPa, we notice that most of the LYP steel dampers are in the yield stage, this conforms to the fact that the LYP steel dampers are the first yield of defense in the steel frame.

From Fig. 36, As can be seen from the stress distribution in the LYP steel dampers, for the triangle load pushover analysis, the maximum stress for the frequent, moderate and rare earthquake levels are 128.02 MPa, 206.26 MPa and 209.41 MPa respectively, for the frequent and moderate earthquake levels we can observe that the LYP steel dampers are mostly in the near yield stage, for the rare earthquake level, we can observe that the LYP steel damper are mostly in the range of collapse and yield stage, we can say the stress in the damper increases as the earthquake level increases and vice versa. We can say that the LYP steel dampers are good energy dissipation agents since they mostly absorb the seismic energy during a seismic occurrence in order to protect the main steel frame from destruction.

Fig. 36Distribution of stress of LYP steel damper under triangle load for frequent, moderate and rare earthquake level

a) Triangle frequent

b) Triangle moderate

c) Triangle rare

We can notice that the difference in the maximum stress of the three earthquake waves and the maximum stress of the LYP steel damper from the pushover analysis under the uniform and triangle loads, the difference in the uniform load pushover analysis is 60, 97 %, the reason is that the braces in the steel frame yielded, so this would not give us an accurate comparison between both the time history analysis and the pushover analysis.

From the pushover analysis under the triangle load, we can notice that the maximum difference in the maximum stress between the time history analysis and the pushover analysis is 20.52 % in the artificial frequent earthquake level, the triangle load pushover analysis would give us a more accurate comparison between both the time history analysis maximum stress and the pushover analysis maximum stress, so from all these observations, we can say that the maximum stress in the damper is almost similar in both results.