Direct displacement-based seismic design of steel eccentrically braced frame structures

Abstract

This paper details a direct displacement-based design procedure for steel eccentrically braced frame (EBF) structures and gauges its performance by examining the non-linear dynamic response of a series of case study EBF structures designed using the procedure. Analytical expressions are developed for the storey drift at yield and for the storey drift capacity of EBFs, recognising that in addition to link beam deformations, the brace and column axial deformations can provide important contributions to storey drift components. Case study design results indicate that the ductility capacity of EBF systems will tend to be relatively low, despite the large local ductility capacity offered by well detailed links. In addition, it is found that while the ductility capacity of EBF systems will tend to reduce with height, this is not necessarily negative for seismic performance since the displacement capacity for taller EBF systems will tend to be large. To gauge the performance of the proposed DBD methodology, analytical models of the case study design solutions are subject to non-linear time-history analyses with a set of spectrum-compatible accelerograms. The average displacements and drifts obtained from the NLTH analyses are shown to align well with design values, confirming that the new methodology could provide an effective tool for the seismic design of EBF systems.

Appendix 1: Illustrating the design procedure for the 5-storey structure

In order to illustrate how the design procedure is applied, this appendix presents the detailed solution for the 5-storey EBF structure. In line with the flowchart of Fig. 9, in addition to the material properties and design criteria identified in Sect. 4.1, one must identify trial section sizes. For this case study practical section sizes are selected as shown in Table 7, with only two different section types initially proposed for columns, braces and links. In addition, desirable link lengths are set, and as shown Table 7, the trial values imply that the links are “short” according to the EC8 classification procedure.

Table 7 Initial section sizes, link lengths and brace angles for the 5-storey EBF

The next step in the design is to compute the yield drift and drift capacity for each storey. In order to do this, the vertical deflection of the beams at yield is first computed using Eq. (8) and then inserted into Eq. (10) to obtain the yield drift component due to shear and bending deformations of the beams and links. The results so obtained are shown in the second and third columns of Table 8. In line with the recommendations of Sects. 3.2.2 and 3.2.3, average strain values of $k_{ br ,i}$ and $k_{, cols ,i-1}$ are assumed for the braces and columns. As shown in Table 8, initial values of 0.25 are assumed for both factors. Eqs. (13) and (14) are then used to compute the yield drift components due to brace and column deformations, respectively. As per Eq. (16), the three drift components are then summed to obtain the total yield drift and the results are shown in Table 8. Subsequently, the plastic drift capacity of each storey is computed by inserting the repairable damage limit state chord rotation for short links from Table 1 into Eq. (18). The plastic drift is then summed, as per Eq. (17), with the yield drift to obtain the total storey drift capacity for structural elements and the results are shown in the last column of Table 8. As the values for structural storey drift capacity are all greater than the storey drift limit of 2.5% for non-structural elements, the structural storey drift limits in Table 8 are identified as critical.

Table 8 Initial estimates of the yield drift and total structural drift capacity for the 5-storey EBF

The minimum storey yield drift from Table 8 is found to be 0.31 % and the minimum drift capacity is 1.22 %. These two values are then inserted into Eq. (19b) in order to obtain the displacement profile for the selected performance limit state, which is shown in Table 9. At this stage a higher mode drift amplification factor should be obtained from Fig. 7c and as the building is only five storeys high, the drift amplification factor is $\omega _{\theta }= 1.0$, such that computation of the final design displacement profile using Eq. (20) leads to the same displacements as the limit state displacement profile. This displacement profile is used to obtain the first-mode design drift profile, $\theta _{i}$, reported in column 4 of Table 9. The ductility demands at each storey are then found using Eq. (23) and the results are shown in the central column of Table 9.

Table 9 Intermediate design results for the 5-storey EBF structure

With the design displacement now known, the substitute structure characteristics should be found using Eqs. (4)–(6). As such, an additional four columns are added onto Table 9 to permit the calculation of the numerators and denominators to be inserted into Eqs. (4)–(6), which leads to the preliminary substitute structure characteristics shown below:

$$\begin{aligned} \Delta _d&= \frac{\sum {m_i \Delta _i^2}}{\sum {m_i \Delta _i}} =\frac{8.6}{76.6}=0.112\,\mathrm{m}\end{aligned}$$

(30)

$$\begin{aligned} m_e&= \frac{\sum {m_i \Delta _i}}{\Delta _d }=\frac{76.6}{0.112}=686.7\,\mathrm{T}\end{aligned}$$

(31)

$$\begin{aligned} H_e&= \frac{\sum {m_i \Delta _i h_i}}{\sum {m_i \Delta _i} }=\frac{925.2}{76.6}=12.07\,\mathrm{m} \end{aligned}$$

(32)

In order to obtain the design base shear for the EBF the only remaining substitute structure characteristic to be computed is the displacement reduction factor via Eq. (21). This in turn requires an estimate of the system ductility demand, which, as explained in Sect. 3.6, requires knowledge of the relative shear proportions over the height of the frame. To establish these, an equivalent lateral force distribution is identified for a unit base shear using Eq. (26) and the subsequent shear profile is computed through Eq. (27). The resulting equivalent lateral force distribution and shear profile are shown in Table 10.

Table 10 Computations used for determination of the system ductility and P-delta component

In line with Eq. (24), the products of the storey shear proportions and design storey drifts are then used to compute the system ductility demand as shown in Eq. (33). The values of the numerator and denominator in Eq. (33) come from the sum of the columns on the right side of Table 10.

$$\begin{aligned} \mu =\frac{\sum \nolimits _{i=1}^{i=n} {\mu _i V_i \theta _i} }{\sum \nolimits _{i=1}^{i=n} {V_i \theta _i}}=\frac{0.0900}{0.0319}=2.82 \end{aligned}$$

(33)

The system ductility is then used to compute the displacement reduction factor from Eq. (21):

$$\begin{aligned} \eta _{\Delta ,EBF} =\left( {1+\frac{1.17\left( {\mu -1} \right) }{1+e^{\mu -1}}} \right) ^{-1}=\left( {1+\frac{1.17\left( {2.82-1} \right) }{1+e^{2.82-1}}} \right) ^{-1}=0.58 \end{aligned}$$

(34)

and the design displacement spectrum is subsequently scaled by 0.58 according to Eq. (22). Using the design displacement of $\Delta _{d}= 0.112\,\mathrm{m}$, the required effective period is then read off the scaled design displacement spectrum (in a manner similar to that shown in Fig. 8b) which leads to a required effective period of $\mathrm{T}_\mathrm{e}=1.49\,\mathrm{s}$. The required effective stiffness is then calculated from Eq. (1) as:

$$\begin{aligned} K_e =4\pi ^{2}\frac{m_e}{T_e^2}=4\pi ^{2}\frac{686.7}{1.49^{2}}=\hbox {12,206}\,\mathrm{kN/m} \end{aligned}$$

(35)

Initially ignoring P-Delta effects, the design base shear (Eq. 2) is then:

$$\begin{aligned} V_b =K_e \Delta _d =\hbox {12,206}\times 0.112=\hbox {1,362}\,\mathrm{kN} \end{aligned}$$

(36)

One can then consider the P-delta stability coefficient for the system:

$$\begin{aligned} \theta _{P\Delta } =\frac{m_e g\Delta _d}{V_b H_e }=\frac{686.7\times 9.81\times 0.112}{\hbox {1,362}\times 12.07}=0.048 \end{aligned}$$

(37)

And as the stability coefficient is less than 0.05, the design base shear does not need to be amplified to account for P-Delta effects using Eq. (25). The design base shear is now distributed as a set of equivalent lateral forces in accordance with Eq. (26) and the design storey shear and the link design shear forces are then found through Eqs. (27) and (28) respectively. The resulting design actions are reported in Table 11.

Table 11 Design shear forces and link resistance ratios for the initial solution of the 5-storey EBF

Table 11 also presents the design shear resistances for the trial link sizes. To obtain these resistance values, the link yield strength is first computed using Eq. (38) from Eurocode 8  [7]:

$$\begin{aligned} V_{y,link} =\left( {{F_y}/{\sqrt{3}}} \right) \cdot t_w \left( {h-t_f} \right) \end{aligned}$$

(38)

where $F_{y}$ is the steel yield strength, $h$ is the link section depth and $t_{w}$ and $t_{f}$ are the web and flange thicknesses respectively. The design strength obtained from DDBD is that required at the development of the design displacement (or limit state). As such, the likely resistance considered during design should be that expected at the same level of deformation demand, with account for hardening if required. In line with this, assuming that the ultimate strength of braces is 1.5 times the yield strength, the design link resistances in Table 11 (and subsequent tables) have been computed as:

$$\begin{aligned} V_{R,link,i}&= \frac{\theta _i}{\theta _{y,i}}V_{y,link,i}\, \quad \mathrm{for}\,\mu <1.0\end{aligned}$$

(39a)

$$\begin{aligned} V_{R,link,i}&= V_{y,link,i} \left( {1+0.5\frac{\gamma _{p,i} }{\gamma _{p,u}}} \right) \quad \mathrm{for}\,\mu \ge 1.0 \end{aligned}$$

(39b)

where $\gamma _{p,u}$ is the ultimate chord rotation capacity of the link (taken here to be 10 % for short links as per Table 1) and $\gamma _{p,i}$ is the expected plastic chord rotation demand which can be estimated from:

$$\begin{aligned} \gamma _{p,i} =\left( {\theta _i -\theta _{y,i}} \right) \frac{L_b }{e_i} \end{aligned}$$

(40)

With the design resistances known, the excess-strength ratios ($V_{R, link }/V_{E, link })$ at each storey are computed and reported in Table 11.

Over the lower three storeys the assumed link sizes are quite close to the design loads, with excess-strength ratios of around 1.2. However, the excess-strength ratios are unacceptably high in the upper storeys, with the top storey value of 2.14 implying that the excess-strength ratios computed for all storeys vary by a factor of close to 2.0, significantly greater than the recommended limit of 1.25 (refer Sect. 3.7).

Given the results presented above, the first change that will be made to the trial design solution is to reduce the section size and length of the links at the top two storeys. The revised link sizes are shown in Table 12.

Table 12 Revised link sizes and estimated drift values for the 5-storey EBF

At this stage the design procedure is repeated, with the main difference being that the storey yield drifts and drift capacities are revised to account for the new link sizes. Table 12 reports the new drift estimates, and note that the column and brace drift components in Table 12 have been computed using the same average strain values ($k_{ br ,i}$ and $k_{ cols ,i-1}$) as in Table 8.

For the revised link sizes the system design displacement remains constant at 0.105 m but the design base shear increases slightly to 1,367 kN owing to the change in ductility demands at the fourth storey. This design base shear is then distributed up the height and the new design shear forces shown in Table 13 are used to check the link strengths, as shown.

The results in Table 13 indicate that the new link sizes are quite good; the excess-strength ratios vary by less than 1.25 over the height of the structure and even though slightly smaller sections might be possible, the sections are considered suitable for this phase of the design.

The next step is to compute capacity design actions, size the columns and braces and then update the average strain values ($k_{br,i}$ and $k_{ cols ,i-1})$ initially assumed in Table 8. For the 5-storey case study structure the capacity design actions are obtained by multiplying the internal actions associated with the link shear resistances by a factor of 1.5, as explained in Sect. 3.8. This leads to the capacity design axial forces shown in Table 14 for the columns and braces. Note that for the columns, the capacity design axial forces shown in Table 14 include axial loads obtained from gravity actions expected during the earthquake.

The initial column and brace sizes shown in Table 7 are checked against the design actions and more efficient section sizes are subsequently selected. Table 14 reports the final column and brace sizes proposed, together with their design values of resistance calculated in line with the standard recommendations provided in Eurocode 3 [38]. With the column and brace sections now established, the average strain values for the braces and columns ($k_{br,i}$ and $k_{ cols ,i-1})$ are computed using Eqs. (12) and (15), respectively. The computed values are compared with initial estimates in Table 15, which also includes new average strain values for the next iteration of design. Note that the rev.1 average strain values proposed for the braces do not exactly match the computed values because as soon as the new average column strain values are imported into the excel design sheet then new average strain values are computed (again via Eq. 12) and these are taken for the brace rev.1 values.

Table 13 Revised design shear forces for the 5-storey EBF

Table 14 Design results for the 5 storey EBF columns and braces

Table 15 Design results for the 5 storey EBF columns and braces

The revised average strain values are used to revise the storey yield drifts and drift capacities (through Eqs. 13, 14, 16 and 17), leading to a design displacement of $\Delta _{d}=0.106\,\mathrm{m}$, which is smaller than the previous values reflecting the fact that the average strain values in the columns and braces tended to be smaller than the initial estimates. The system ductility is then recomputed as before and is also found to reduce, to a value of $\mu =2.58$. A new effective period is determined; $T_{e}=1.36\,\mathrm{s}$, which leads to a slightly increased design base shear of $V_{b}=\hbox {1,543}\,\mathrm{kN}$. New design actions for the links, braces and columns are subsequently found and the average strain values shown in Table 15 as rev.1 are obtained. The results in Table 15 show that reasonably good convergence has been obtained after just a single iteration. However, by undertaking an additional iteration using estimated rev.2 values set equal to the calculated rev.1 values in Table 15, convergence to within 1 % is obtained at all floors and the final design base shear is found to be $V_{b}=\hbox {1,495}\,\mathrm{kN}$. Distributing this value over the height of the building, the link sizes are checked again in Table 16 and it can be seen that the link sizes are acceptable, with excess-strength ratios that vary by no more than 10 %. In addition, capacity design axial forces in the braces and columns are checked against resistances (see right side of Table 16) and it is confirmed that the section sizes previously listed in Table 14 are acceptable.

Table 16 Final design checks for the 5-storey EBF

At this stage the DBD procedure and member sizing is complete. Only the design of connections would remain but in this work, as all connections are assumed to be pinned, their detailed design has not been undertaken insofar as the connection details are not required for the non-linear time-history analysis models used to verify the procedure in Sect. 4.5.