# Shear strength and stiffness enhancement of cross-stiffened steel plate shear walls

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## Abstract

Nowadays, the use of steel plate shear walls, as an effective seismic resisting system, has been of great interest in enhancing the lateral strength and stiffness of buildings both in renovation and seismic rehabilitation of existing concrete and steel structures. In the present research, the shear strength and stiffness of steel plate shear walls in various configurations of stiffeners, including horizontal, vertical, and horizontal–vertical, were investigated by finite element method and finally semi-empirical relations were presented in this regard. The results indicated that the shear strength and stiffness of stiffened SPSWs were well predicted by the proposed relations, but increasing the number of stiffeners above a certain range will not have a significant effect on enhancing the stiffness and strength.

## Keywords

Stiffened steel plate shear wall Effective shear yield strength Stiffness Finite element analysis Stiffener## Introduction

In recent years, extensive experimental investigations have been conducted under cyclic and monotonic loads on steel plate shear walls (SPSWs) to ensure the seismic performance of the system. The results have revealed high stiffness, sufficient strength, excellent ductility, and high energy absorption of this seismic resisting system. The researchers have been analytically interested in steel plate shear walls as the steel plate shear walls are used in seismic rehabilitation of existing structures in addition to newly constructed structures.

Alinia and Dastfan compared the behavior of unstiffened (thin) and stiffened panels in 2006 and 2007. In this numerical study, the effect of stiffening was investigated on the ultimate strength and cyclic behavior of stiffened and unstiffened panels. They found that the optimal number of stiffeners should be used to achieve sufficient rigidity and ductility (Alinia and Dastfan 2006, 2007). Sabouri ghomi et al. conducted studies to improve the stability and prevent the early elastic buckling of the wall using horizontal and vertical stiffeners. They also observed that the proper application of stiffeners would enhance the energy absorption and stiffness of the system (Sabouri-Ghomi et al. 2008).

Alinia and Sarraf Shirazi provided a practical design method for stiffening thin SPSWs (Alinia and Sarraf Shirazi 2009). Sabouri-Ghomi and Sajjadi proposed a method to determine the minimum moment of inertia for stiffeners to prevent the global buckling of the plate (Sabouri-Ghomi and Sajjadi 2012). The values resulting from this method were compared with the tests conducted by Takahashi et al. (1973) where a good agreement was reached.

They also studied the performance of unstiffened panel between two openings in stiffened steel walls by finite element methods. The results of their research suggested that there will be limitations on the dimensions of the cross-section of stiffener, as well as the height–width ratio of intermediate panel such that intermediate thin steel plate is yielded sooner than the dual stiffeners on both sides (Sabouri et al. 2013).

Nie et al. recommended a design method for calculating the lateral resistance capacity of stiffened SPSWs based on the experimental research (Nie et al. 2013). Machaly et al. numerically investigated the ultimate shear strength of SPSWs (Machaly et al. 2014). Brando and De Matteis provided design curves for low strength-high hardening metal multi-stiffened shear plates, based on both experimental tests and parametric numerical analyses (Brando and De Matteis 2014). Sabouri-Ghomi and Mamazizi conducted an experimental investigation of stiffened SPSWs with two rectangular openings (Sabouri-Ghomi and Mamazizi 2015).

Zirakian and Zhang studied the buckling and yielding behavior of unstiffened slender, moderate and stocky low yield point SPSWs with various support and loading conditions (Zirakian and Zhang 2015). Guo et al. explored the influence of hinged, rigid, and semi-rigid connection joints on the behavior of stiffened and unstiffened SPSWs using experimental test and finite element analysis (Guo et al. 2015). Rahmzadeh et al. described the effect of the rigidity and arrangement of stiffeners on the buckling behavior of plates (Rahmzadeh et al. 2016). Jin et al. investigated the stability of buckling-restrained SPSWs with inclined-slots (Jin et al. 2016). Guo et al. analyzed the failure mode, energy dissipation mechanism, and the influence of joint connecting forms and arrangement of stiffeners on the seismic performance of cross-stiffened SPSWs with a semi-rigid connected frame (Guo et al. 2017).

Haddad et al. experimentally investigated the cyclic performance of stiffened steel plate shear walls with three configurations including cross-stiffened, circular-stiffened, and diagonally stiffened SPSWs (Haddad et al. 2018). Afshari and Gholhaki proposed an equation for shear strength degradation of unstiffened steel plate shear walls with optional located opening (Afshari and Gholhaki 2018).

Despite many studies performed on the stiffened steel plate shear walls by various researchers, these walls require time-consuming and expensive non-linear static and dynamic analyses to capture the strength and stiffness of the system. On the other hand, almost all classical and theoretical relations associate with unstiffened steel plate shear walls, and there is no definite relation to estimate the strength and stiffness of the stiffened panel compared to the unstiffened panel. Accordingly, the purpose of this paper is to achieve simple relations computing the strength and stiffness of the stiffened panel based on the number of vertical and horizontal stiffeners converting the global buckling mode of the plate to the local buckling mode in sub-panels.

## Validation of mesh sensitivity and modeling

Results of mesh sensitivity analysis

Finite element mesh Size (mm) | Average error in relation to experimental test (%) | Analysis time (min) |
---|---|---|

150 × 150 | 13 | 15 |

100 × 100 | 6.3 | 35 |

60 × 60 | 2.1 | 45 |

30 × 30 | 2 | 195 |

15 × 15 | 1.2 | 455 |

Considering the aspect ratio of the shear panels of the present study with the above-mentioned experimental test and given the scale of the models of the present research, the 150 mm by 150 mm mesh size was selected for all of the subsequent modellings.

Mechanical properties of steel material used in the finite element modeling

Steel material | Material type | Young’s modulus (MPa) | Yield strain (%) | Hardening strain (%) |
---|---|---|---|---|

Infill plate | ST14 | 180E3 | 0.107 | 2.78 |

Frame | ST52 | 210E3 | 0.197 | 3.46 |

Stiffener | ST37 | 205.94E3 | 0.116 | 3.27 |

## Modeling of studied specimens

To achieve simple relations representing the increase in the percentage of strength and stiffness of the stiffened panel with stiffener compared to the same model without stiffener, a panel without stiffener should first be modeled. Concerning common bay length of buildings, a steel plate shear wall 5000 mm wide, 3000 mm high and 2.5 mm thick was considered with a simple connection surrounding frame. Stiffness of the beam and column is such that the plate is completely under pure shear.

To select proper sections for the surrounding elements with sufficient rigidity, the specifications of AISC341-16 (AISC 2016a) were used. The preliminary design of surrounding columns and the connecting beam, given its simple connections to adjacent columns, was performed using Eqs. (1) and (2), respectively (AISC 2016b).

*I*

_{c},

*t*

_{w},

*d*,

*b,*and

*M*

_{pb}are column moment of inertia, plate thickness, plate height, plate width, and plastic moment capacity of the beam, respectively.

*α*represents the diagonal tension field angle as shown in Relation (3). In addition, the stress of tension field at the yielding time (

*σ*

_{ty}) is obtained by solving Eq. (4) (AISC 2016b).

*A*

_{c},

*A*

_{b}and

*τ*

_{cr}denote the column cross-sectional area, beam cross-sectional area, and the critical buckling shear stress of steel plate, respectively.

*τ*

_{cr}can be determined in accordance with classical stability Equation (Timoshenko and Gere 1961) as follows:

*ν*,

*E*,

*τ*

_{y}, and

*F*

_{y}are Poisson’s ratio (0.3), elasticity modulus, shear yielding stress, and uniaxial tensile yielding stress of the plate, respectively. The shear buckling factor

*k*which depends on the steel plate aspect ratio and boundary conditions, which is equal to 18.861 for the simple supported plate of the present research. Accordingly, for the wall with the mentioned sizes, beams and columns of type IPB300 were designed and modeled as displayed in Fig. 6.

*τ*

_{cr}(0.878 MPa ≈ 0) obtained from Eq. (5),

*σ*

_{ty}is almost equal to specified minimum yield stress (uniaxial yield stress,

*F*

_{y}) of infill plate. Hence, the shear strength of the steel plate shear wall is obtained by Relation (6).

Concerning the value of *α* (42.7°) which was obtained by Eq. (3), theoretical shear strength of the wall was 1466.27 kN. Based on finite element model analysis of unstiffened panel, shear strength of the plate was 1425.89 kN, indicating 2.8% error compared to analytical value of 1466.27 kN. This demonstrates the proper accuracy of mesh sizes and modeling process.

## An introduction to arrangement and dimensions of stiffeners

To convert the global buckling of the plate to the local buckling mode in each sub-panel, the relation provided by Timoshenko for orthotropic plates must be satisfied in accordance with Relation (7) (Timoshenko and Gere 1961).

*k*

_{G}is the global buckling coefficient of the plate, calculated as 3.64 in terms of the simple connection of the plate to the surrounding members, and

*k*

_{L}is the local buckling coefficient of the plate obtained from Relation (8).

*I*

_{x}=

*I*

_{y}), and by replacing

*I*

_{s}with

*I*

_{x}and

*I*

_{y}in Relation (7), and with respect to

*S*

_{x}and

*S*

_{y}values obtained from arrangements of stiffeners based on Fig. 7, the moment of inertia value (

*I*

_{s}) is calculated for the horizontal and vertical stiffeners of the examined models, with the highest value of (30.16 × 10

^{4}mm

^{4}) for the model 0v4h. Therefore, by choosing stiffeners with 10 mm width (

*b*

_{s}= 100 mm) and 10 mm thickness (

*t*

_{s}= 10 mm), according to Relation (9), the moment of inertia corresponding to the chosen dimensions is equal to 33.33 × 10

^{5}mm

^{4}which is higher than the highest value required for the 0v4h model. Therefore, the moment of inertia required for stiffeners of all models is provided to supply the local buckling of sub-panels.

In the above relation, *F*_{ys} is the yield stress of stiffening materials.

### Analysis and study of results

In Fig. 10, *V*_{y} is the effective shear yield strength (shear strength), *V*_{u}, is the ultimate lateral load carrying capacity of the shear panel, *K*_{e} represents the effective shear stiffness, *δ*_{y} shows the limiting elastic shear displacement, and *δ*_{u} is displacement at the moment of the ultimate load carrying capacity.

### Analysis of the displacement–force curves obtained from model’s analysis

*V*

_{u}), the limiting elastic strength (

*V*

_{y}), and the panel stiffness (

*K*

_{e}) are expected to increase, while the ultimate shear displacement of the structure (

*δ*

_{u}) is expected to diminish due to adding the stiffener to the shear panel. Figures 11, 12, 13, 14 and 15 present the effect of enhancing the strength and stiffness and reducing ultimate displacement of the structure after addition of stiffeners. In these figures, changes in shear panel force–displacement curve without vertical stiffeners are observed after adding only the horizontal stiffener from zero to four stiffeners (Fig. 11). The process of adding horizontal stiffener from 0 to 4 for a panel with one vertical stiffener up to 4 vertical stiffeners is demonstrated in Figs. 12, 13, 14 and 15, respectively.

Effective shear yield strength of the studied models

Vy (kN) | 0v | 1v | 2v | 3v | 4v |
---|---|---|---|---|---|

0h | 1425.89 | 1486.70 | 1551.42 | 1583.78 | 1618.11 |

1h | 1479.83 | 1539.65 | 1592.61 | 1631.83 | 1667.14 |

2h | 1552.40 | 1609.28 | 1661.25 | 1698.52 | 1729.90 |

3h | 1581.82 | 1642.62 | 1695.58 | 1729.90 | 1763.24 |

4h | 1653.41 | 1712.25 | 1764.23 | 1799.53 | 1832.87 |

Effective shear stiffness of the studied models

Ke (kN/m) | 0v | 1v | 2v | 3v | 4v |
---|---|---|---|---|---|

0h | 120,720.5 | 156,220.7 | 190,838.4 | 222,121.8 | 250,757.3 |

1h | 157,299.5 | 188,288.6 | 216,728.1 | 243,304.2 | 267,624.8 |

2h | 216,728.1 | 237,322.1 | 256,935.5 | 275,568.3 | 293,220.3 |

3h | 291,749.3 | 303,027.0 | 312,931.8 | 322,738.5 | 332,251.0 |

4h | 346,568.8 | 351,962.5 | 354,021.9 | 355,983.2 | 357,846.5 |

### Determination of semi-empirical relations for estimating the model’s strength

*n*

_{h}) and vertical (

*n*

_{v}) stiffeners added to the shear panel. Accordingly, Fig. 17 shows the average percentage of variations in the strength of the models compared to unstiffened thin panel for two modes of increase in vertical stiffeners (

*n*

_{v}), regardless of the number of horizontal stiffeners as well as the increase in the horizontal stiffener (

*n*

_{h}), irrespective of the number of vertical stiffeners. The best curves passing through each of data categories with different equations are fitted in two precise and approximate states according to Fig. 17. Finally, Relation (11) presents the percentage of changes in the strength based on the number of horizontal and vertical stiffeners.

*ΔV*

_{y}is the percentage increase of the shear strength of the stiffened panel compared to unstiffened panel. To simplify the above relation, we used power fitness as depicted in Fig. 17 to obtain a simple as Relation (12).

### Determination of semi-empirical relation for estimating the model’s stiffness

This trend in (1h) is also greater than in (2h) and the aforementioned course decreases after adding the number of horizontal stiffeners. Therefore, unlike the trend mentioned for the strength of specimens, use of mean of variations in this case will yield a considerable error. Therefore, to minimize the calculation error and provide a simple empirical relation for increasing the stiffness of specimens, it is necessary to use the fitting functions (B and C) of Fig. 18b, c respectively in relation to the coefficients along with the power of resulting equations of Fig. 18a.

Accordingly, an exact relation suggesting the percentage of increase in the stiffness due to the effect of adding vertical stiffeners is achieved separately for any particular number of horizontal stiffeners while not using the average variation. The above process takes into account the trend of increase of stiffness of the values of the row in Table 4 in the given relation.

*ΔK*_{e} is the increasing percentage of shear stiffness of stiffened panel compared to the unstiffened panel. Another remarkable point in this section, as seen in Table 4, is the greater effect of horizontal stiffener on the elevation of stiffness of stiffened panel compared to vertical stiffener. Comparing any arbitrary couples of models of the studied panels whose total stiffeners are equal, but the number of their horizontal and vertical stiffeners are exactly opposite, it is observed that a model with a higher number of horizontal stiffeners will have greater stiffness. The reason may be the ratio of (*S*_{y}/*S*_{x}), which ultimately leads to a more uniform tension field between the beams. The above issue is not the case for increased strength of the panels, and the effect of horizontal and vertical stiffeners on the panel’s strength rise is almost the same. In other words, if the goal is only to enhance the stiffness of the panel with one stiffener, then this stiffener should be used horizontally.

### Verification of the proposed relation

Prediction of models’ strengths using the proposed Relation (11)

| 0v | 1v | 2v | 3v | 4v | 5v | 6v | 7v |
---|---|---|---|---|---|---|---|---|

0h | 1425.89 | 1486.31 | 1539.49 | 1579.06 | 1608.47 | 1630.77 | 1648.07 | 1661.78 |

1h | 1471.93 | 1532.35 | 1585.53 | 1625.10 | 1654.51 | 1676.81 | 1694.11 | 1707.82 |

2h | 1537.37 | 1597.79 | 1650.97 | 1690.54 | 1719.95 | 1742.25 | 1759.55 | 1773.26 |

3h | 1593.77 | 1654.19 | 1707.36 | 1746.93 | 1776.34 | 1798.65 | 1815.94 | 1829.65 |

4h | 1637.08 | 1697.50 | 1750.68 | 1790.24 | 1819.66 | 1841.96 | 1859.26 | 1872.96 |

5h | 1669.44 | 1729.86 | 1783.04 | 1822.61 | 1852.02 | 1874.32 | 1891.62 | 1905.33 |

6h | 1693.69 | 1754.11 | 1807.29 | 1846.85 | 1876.27 | 1898.57 | 1915.87 | 1929.57 |

7h | 1712.11 | 1772.53 | 1825.71 | 1865.27 | 1894.69 | 1916.99 | 1934.29 | 1947.99 |

Prediction of models’ stiffness using the proposed Relation (13)

Ke (kN/m) | 0v | 1v | 2v | 3v | 4v | 5v | 6v | 7v |
---|---|---|---|---|---|---|---|---|

0h | 129,984.0 | 166,768.6 | 198,669.5 | 228,954.5 | 258,235.7 | 286,792.5 | 314,784.9 | 342,316.5 |

1h | 153,805.2 | 184,869.6 | 211,796.0 | 237,354.0 | 262,062.0 | 286,156.7 | 309,773.6 | 333,000.4 |

2h | 214,429.2 | 236,263.7 | 254,820.4 | 272,312.7 | 289,147.9 | 305,510.7 | 321,506.7 | 337,203.9 |

3h | 294,573.4 | 306,652.3 | 315,540.4 | 323,522.9 | 330,968.7 | 338,039.4 | 344,824.9 | 351,381.9 |

4h | 342,234.9 | 347,689.4 | 350,264.4 | 352,302.6 | 354,055.2 | 355,622.3 | 357,055.6 | 358,386.6 |

5h | 358,644.5 | 360,834.4 | 361,336.6 | 361,682.2 | 361954.0 | 362,181.5 | 362,378.9 | 362,554.4 |

6h | 363,162.9 | 363,996.5 | 364,082.5 | 364,136.9 | 364,177.4 | 364,210.0 | 364,237.4 | 364,261.1 |

7h | 364,326.7 | 364,637.4 | 364,652.4 | 364,661.5 | 364,668.2 | 364,673.4 | 364,677.8 | 364,681.5 |

The proper congruence between the results of the finite element modeling for the validation specimens and the values of the proposed relation confirm the validity of the relations. The results suggest that maximum 7 horizontal and vertical stiffeners provide the maximum strength and stiffness of the panel. This specific bound is exactly the boundary, after which the diagonal tension field will not completely form in all sub-panels.

By comparing the shear strength values given in Table 5 obtained using the proposed Relation (11), with the corresponding values included in Figs. 19, 20, 21 and 22 derived from the finite element modeling of the verification models, it can be observed that the error rate due to application of the proposed Relation (11) instead of the time-consuming finite element analysis for verification models 5v0h, 0v5h, 1v5h, and 4v6h, was 0.30, 0.76, 0.43, and 0.51%, respectively. This indicates the high accuracy of the proposed Relation (11) in the estimation of shear strength of stiffened panels. Similarly, the comparison of the predicted stiffness values for the four above-mentioned validation models (Table 6) and the corresponding values obtained from finite element analysis indicated an error rate of 3.05, 0.61, 0.39, and 0.55%, respectively, due to applying the proposed Relation (13). This clearly confirms the accuracy of the proposed relation in the estimation of the stiffness of the stiffened models.

Verification of the proposed Relation (11) with experimental tests

The results clearly indicate the high accuracy of the proposed relations regarding the increasing rate in the strength of the stiffened shear panel with the arbitrary number of horizontal or vertical stiffeners.

## Conclusions

The results indicated that the unstiffened thin steel plate shear wall possessed a high ductility. Also, the lateral load-carrying capacity of the wall increased due to adding stiffeners and limiting different displacements and out-of-plane deformations, as well as the enhanced stiffness of the structure. The results indicated that there is a special upper bound for the number of horizontal and vertical stiffeners in the stiffening process of a specific panel. Addition of stiffeners beyond the allowed value had no significant effect on enhancing the stiffness and strength of the panel, and only increased the weight of the panel and executive problems. Investigations of the present paper introduced the number 7 for horizontal and vertical stiffeners, as the valid range for using the proposed relation for the percentage increase in strength and stiffness. The results revealed that horizontal stiffener was more effective in enhancing the stiffness of the panel than vertical stiffener was. However, the effect of adding whether horizontal and vertical stiffeners on the strength rise was almost the same. In other words, if the goal is to enhance the stiffness of the panel with a stiffener, this should be horizontal. The results also suggested that achieving simple relations is possible to calculate the stiffness and strength of a stiffened panel from an unstiffened panel. Comparison of the error rate obtained using the simplified relations of the percentage increase in strength (Relation 12) instead of the corresponding exact relation (Relations 11) indicated that the use of the simplified relation was very useful where the recorded error was negligible (less than 2.65%) in all models.

## Notes

### Compliance with ethical standards

### Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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