A variable stiffness energy dissipation device for drift control of steel frames

Abstract

A novel variable stiffness energy dissipation device is conceptualized and numerically studied in this paper. The proposed device is a passive device that consists of an array of helical springs arranged in the form of a series of von Mises trusses anchored between two tubes. Under an external excitation, the tubes move relative to each other and energy is dissipated when the von Mises trusses undergo snap-through and snap-back actions. After presenting the theoretical underpinning of this device, its behavior is programmed using C++ and incorporated into the framework of an open-source software OpenSees. Non-linear time history analyses are then performed on several shear, moment resisting and braced frames to demonstrate the effectiveness of the device in dissipating energy and reducing structural deformations. The results show that the proposed device is capable of reducing both the peak and residual displacements of shear and moment frames, and preventing brace buckling of braced frames under three different levels of earthquake severity.

Keywords

energy dissipation snap-through instability earthquake resistant design vibration control steel frames

Introduction

Performance-based Seismic Design (PBSD) is a design concept currently used for the seismic design of buildings and bridges (FEMA, 1996). It allows building owners to define performance levels to meet certain specific requirements for the building and its contents. The main goal of PBSD is to design structures to exhibit predictable behaviors under an earthquake (Qammer and Dalal, 2016). Such methodology allows engineers to determine the appropriate level of ground motion that should be applied to the structure and decide on the level of protection that particular structure needs for that ground motion.

Figure 1 shows four standard performance levels for three building groups under three levels of earthquake. Group I to III buildings represent structures that require an increasing level of protection, from basic, to essential, to safety critical facilities. The primary goal of selecting the proper performance objectives is to control or limit the risk resulting from earthquake effects. For design purposes, a building’s transient and permanent drifts are often used as performance indices to determine whether it satisfies its design performance objective. For moment resisting and braced steel frames, the limits placed on transient and permanent drifts are given in Table 1 (FEMA, 2000).

Figure 1.

Performance objectives.

Table 1.

Transient and permanent drift limits.

Frame type	Immediate occupancy	Life safety	Collapse prevention
Moment resisting frames	0.7% transient; negligible permanent	2.5% transient; 1% permanent	5% transient; 5% permanent
Braced frames	0.5% transient; negligible permanent	1.5% transient; 0.5% permanent	2% transient; 2% permanent

The frame types shown in Table 1 - Moment resisting and braced frames - are two most frequently used steel frame systems in the US to resist gravity and lateral loads. One advantage of braced frames is they have a relatively high lateral stiffness to limit the amount of drifts. However, under strong seismic excitations, buckling of compression brace member could occur. During a strong earthquake, a large amount of energy is imparted to the structure. If this energy is not dissipated properly by a supplementary damping system, large strains that result in inelastic deformations are likely to occur and they may cause severe structural damage or even collapse of the building. To reduce strains and deformations in the primary structural members, it is necessary to introduce a means by which this input energy is safely and effectively dissipated. Connor (2003) summarized several external and internal mechanisms by which energy can be absorbed and dissipated, such as dissipation by fluid viscosity, by material viscoelasticity, by friction, and by material undergoing cyclic inelasticity.

Over the years, researchers have developed and used various types of energy dissipation devices on buildings. The use of snap-through instability to dissipate energy and control vibration has gained momentum in recent years. Snap-through instability is a type of buckling that occurs when a system with certain configuration under a specific load pattern suddenly switches from one equilibrium position to another (Sano and Wada, 2018; Sonnerlind, 2014). The theoretically formulation of snap-though buckling was given by Koiter (1945). Since then, a number of papers on the general theory and applications of snap-through instability have emerged. Huang and Vahidi (1971) analyzed the snap-through buckling of two simple structures subjected to quasi-static loading using elastic theory of prismatic bars. Pecknold et al. (1985) identified and clarified some basic characteristics associated with snap-through, bifurcation and post-buckling behavior of structures. Tsai and Palazotto (1991) investigated the large-rotation snap-through behavior of laminated cylindrical panels using nonlinear finite element analysis. Hrinda (2010) analyzed several geometrically nonlinear truss structures with snap-through behavior using an arc length approach implemented in a finite element analysis framework. Zhang et al. (2021) investigated the translational and rotational states of metastructures made from planar arrangements of bi-stable beams undergoing snap-through.

While the theory behind snap-through buckling is well established, the use of snap-through/snap-back buckling as a means to dissipate energy is relatively new. Avramov and Mikhlin (2004) investigated the use of a low mass snap-through truss for elastic oscillation absorption. Ha et al. (2018) developed an energy absorption lattice made from multiple units of tetra-beam-plate cells with negative stiffness. Energy absorption results when the constituent beams of these cells under snap-through instability. Zhao et al. (2020) and Zhang et al. (2020) proposed the use of notched axially-loaded strips arranged in certain patterns to trigger predictable directional snap-through as a means to trap and release energy. Li et al. (2020) developed a continuous laminated metal module of a preloaded beam bimorph and elastic layer to achieve the desired damping properties based on an internal snap-buckling mechanism. To further explore how snap-through instability can be harnessed to help reduce structural vibrations, the main objective of this paper is to propose a device made from helical springs that is capable of dissipating energy when these springs undergo snap-through and snap-back under cyclic loads.

In the following section, the von Mises truss that undergoes snap-through and snap-back buckling under cyclic loading condition is reviewed. The proposed energy dissipation device is then introduced. After studying the parameters that affect the behavior of this device, the implementation of the behavior of the device in OpenSees is discussed. This is followed by a series of nonlinear time-history analyses to demonstrate how the proposed device can be used to reduce transient and permanent deflections of steel frames.

Von mises trusses

As shown in Figure 2, a von Mises truss is a shallow truss structure prone to exhibiting nonlinear snap-through behavior (Kala and Kalina, 2016; Psotny and Ravinger, 2003; Savi and Nogueira, 2010; Snirc and Ravinger, 2017; Wiebe et al., 2011) when subject to the applied load. When the applied load increases, the top joint of the truss deflects down, inducing compressive force in the two truss members. This compression force increases as the applied load increases, and when the two members are in a horizontal position, their theoretical compression force becomes infinite. This is, however, not achievable for a real structure. In a real structure, the upright configuration of the truss becomes unstable when the applied load approaches point A in Figure 2. Once Point A is reached, the truss suddenly snaps to Point B and changes to an inverted position. In the new configuration, the two truss members are in tension. This new configuration is a stable configuration and is thus the preferred equilibrium position for the truss. Any load applied beyond Point B will now follow curve BC. If the load is reversed, the load-deflection behavior follows path CBDE. Snap-back occurs at Point E towards Point F, and the cycle repeats if the load is reversed again. Note that the equilibrium path shown as the dashed line from Point A to Point E will not be realized unless the truss is under a displacement-controlled load condition. The area traced by Points O-A-B-D-E-F-O represents energy that is dissipated in a one cycle of loading, unloading and reloading (Deng et al., 2020). It is also of interest to note that the stiffness of this truss (obtained as the slope of the load-deflection curve) is not a constant. Its value changes with the configuration of the truss.

Figure 2.

Load-deflection behavior of a von mises truss.

Proposed energy dissipation device

The proposed energy dissipation device shown in Figure 3 is constructed using an array of helical springs arranged in the form of a series of von Mises trusses. The device is to be installed between the top of an inverted chevron brace and the beam above it. This arrangement enables the largest relative displacement to occur between the top of the brace and the beam (Okazaki et al., 2013; Ozbulut and Hurlebaus, 2011), thus allowing the device to dissipate the maximum amount of energy.

Figure 3.

Conceptual design of the energy dissipation device.

As shown in Figure 3, one end of a spring is anchored to the inside wall of an outer tube (labelled 6) and the other end is connected to the outside wall of an inner tube (labelled 5). A rigid bar (labelled 7) with stationary bushings (labelled 1–4) is connected to the beam and is allowed to slide inside the inner tube. The bushings are not attached to the inner tubes and the gaps between the bushings and the ends of the inner tubes are present to allow the springs to undergo snap-through and snap-back freely without interference. These bushings are also positioned in such a way that only one set of springs (the left or the right) will undergo snap-through or snap-back instability when the chevron brace (which is connected to the outer tube) and the beam (which is connected to the rigid bar) move relative to each other.

The operating mechanism of the device is depicted schematically in Figure 4. If the excitation causes the rigid bar to move from its at rest position to the right relative to the outer tube (i.e., when the load goes from Point O to Point A), bushing 3 will make contact with the left end of the right inner tube and exert a pushing force on the right set of springs to allow them to undergo snap-through at Point A (i.e., snaps from Point A to Point B), while the left set of spring remains undisturbed. After snap-through occurs, the inclined direction of the right set of springs will be reversed. Bushing 1 is now in contact with the left end of the left inner tube. If the direction of relative movement remains unchanged, the device will follow load path BC. However, if the direction of relative movement reverses, the device will follow load path BD. Bushing 4 will now push on the right end of the right inner tube causing them to undergo snap-back at Point E (i.e., snaps from Point E to Point F). After snap-back, bushing 2 is now in contact with the right end of the left inner tube. If the direction of relative movement remains unchanged, the device will follow load path FG. However, if the direction of relative moment reverses, the device will follow load path FO. Thus, completing one full cycle of snap-through and snap-back. It should be noted that both snap-through and snap-back are instantaneous processes as the system transitions from one stable equilibrium position to another stable equilibrium position, and since bushing 4 is not in contact with the right end of the inner tube during snap-through and bushing 3 is not in contact with the left end of the inner tube during snap-back, snap-through and snap-back will proceed even if the direction of movement is suddenly reversed during the process.

Figure 4.

Snap-through and snap-back experienced by the right set of springs (The arrow indicates relative movement of the rigid bar with respect to the outer tube).

On the other hand, if the excitation causes the rigid bar to move to the left relative to the outer tube, the left set of springs will now be engaged and undergo the snap-through and snap-back processes in a manner similar to the right set of springs. Thus, regardless of the direction of relative movement between the rigid bar and the outer tube, one set of springs will undergo snap-through and snap-back in a cycle of loading and unloading. As indicated earlier, the area enclosed by Points O-A-B-D-E-F-O represents the amount of energy dissipated by the device in a full load cycle. Energy is dissipated when strain energy stored in the truss members under compression is released as irrecoverable kinetic energy when snap-through/snap-back occurs.

Governing equations

A basic component of the proposed energy dissipation device is shown Figure 5. The two identical helical springs are connected at the top and will deflect down by a distance of $u$ under the applied load P. For a given spring, its stiffness ( $k_{s}$ ), free length ( $L_{f}$ ) and the initial inclination angle ( $α_{0}$ ) are all known, or can be adjusted.

Figure 5.

A component of the proposed device.

As the top joint moves down by a distance of $u$ , the new inclination angle is now $α$ , and the new length of the spring becomes $L_{f} + ∆ x$ , where $L_{f}$ is the free length of the spring and $∆ x$ is the change in length of the helical spring. Under the external load, the change in spring length is

∆ x = \sqrt{{(L_{f} \sin (α_{0}) - u)}^{2} + {(L_{f} \cos (α_{0}))}^{2}} - L_{f}

(1)

Summing force vertically, and denoting upward as the positive direction, we have

- P + 2 k_{s} (∆ x) \sin (α) = 0

(2)

Since

\tan (α) = \frac{L_{f} \sin (α_{0}) - u}{L_{f} \cos (α_{0})}

(3)

Substituting Equations (1) and (3) into equation (2), the load (P) versus displacement (u) relationship of the component can be expressed as

P = 2 k_{s} (u - L_{f} \sin (α_{0}) + L_{f} \times \frac{L_{f} \sin (α_{0}) - u}{\sqrt{{(L_{f} \sin (α_{0}) - u)}^{2} + {(L_{f} \cos (α_{0}))}^{2}}})

(4)

Taking derivative of equation (4) with respect to u, we have

\frac{d P}{d u} = 2 N_{s} k_{s} L (\frac{1}{L} - \frac{L_{f}}{L^{2} \sqrt{{(\frac{L_{f}}{L} \sin (α_{0}) - \frac{u}{L})}^{2} + {(\frac{L_{f}}{L} \cos (α_{0}))}^{2}}} + \frac{L_{f} {(\frac{L_{f}}{L} \sin (α_{0}) - \frac{u}{L})}^{2}}{L^{2} {({(\frac{L_{f}}{L} \sin (α_{0}) - \frac{u}{L})}^{2} + {(\frac{L_{f}}{L} \cos (α_{0}))}^{2})}^{\frac{3}{2}}})

(5)

The above equation represents the stiffness of the spring component, which is nonlinear. The points at which snap-through (Point A) and snap-back (Point E) occur can be determined by setting the above equation equal to zero. Once these points are reached, the force-displacement relationship is represented by a horizontal line, and thus the stiffness of the component is zero in regions of snap-through and snap-back. However, it should be noted that while the device exhibits zero stiffness in the snap-through and snap-back regions, the structure itself is not a zero-stiffness structure (Schenk and Guest, 2013; Tarnai, 2003) since the device is just one component of the structure. To locate Point B (for snap-through) and Point F (for snap-back), the displacements that define these points are obtained by solving for u from equation (4) at the P value that corresponds to Point A and Point E, respectively, and enforcing the condition dP/du ≠ 0.

Implementation into opensees

In the present study, OpenSees was used as a platform to perform dynamic analysis. OpenSees (Open Systems for Earthquake Engineering Simulation) is an open-source software developed by the Pacific Earthquake Engineering Research (PEER) Center to address research needs for dynamic analyses and simulations.

As a finite element based object-oriented software framework for earthquake engineering research, most of its modules can be developed on an open-source platform and implemented using a programming language such as C++. The open-source nature of the source code makes it possible for a variety of advanced applications to be implemented. The tool command language (TCL) is the programming language used to define, construct, and execute the analysis.

Due to the complexity of the proposed energy dissipation device, instead of creating a structural model that consists of a series of springs anchored between two tubes with a moving rigid rod, it is much easier to derive the constitutive equation that describes the behavior of the device and implement it directly into OpenSees as a new material. This new material can be used to simulate the uniaxial stress-strain behavior of the entire device. To add a new material to the OpenSees framework, an important task is to define the stress-strain (or constitutive) behavior of the material using C++ and build it into the local library. After the new material is successfully added, it can be assigned to any member in the structural mode. In the present research, the behavior of the proposed device is simulated by the “pseudo” stress-strain behavior of a material and built into the framework.

To achieve the above goal, two additional parameters are introduced. They are the area ( $A$ ) of the member to be assigned or the cross-sectional area of the device, and the length ( $L$ ) of the member to be assigned or the actual length of the device. Since the proposed device consists of a series of inclined spring components shown in Figure 5 mounted between an outer and an inner tube, one other parameter - the number of spring components to be used in the device ( $N_{s}$ ) is also introduced.

After introducing $A$ , $L$ and $N_{s}$ , equation (4) can be rewritten as

\frac{P}{A} = \frac{2 N_{s} k_{s} L}{A} (\frac{u}{L} - \frac{L_{f}}{L} \sin (α_{0}) + \frac{L_{f}}{L} \times \frac{\frac{L_{f}}{L} \sin (α_{0}) - \frac{u}{L}}{\sqrt{{(\frac{L_{f}}{L} \sin (α_{0}) - \frac{u}{L})}^{2} + {(\frac{L_{f}}{L} \sin (α_{0}))}^{2}}})

(6)

σ = \frac{2 N_{s} k_{s} L}{A} (ε - \frac{L_{f}}{L} \sin (α_{0}) + \frac{L_{f}}{L} \times \frac{\frac{L_{f}}{L} \sin (α_{0}) - ε}{\sqrt{{(\frac{L_{f}}{L} \sin (α_{0}) - ε)}^{2} + {(\frac{L_{f}}{L} \cos (α_{0}))}^{2}}})

(7)

The pseudo modulus of the material representing this device can be obtained by taking derivative of equation (7) with respect to ε.

\frac{d σ}{d ε} = \frac{2 N_{s} k_{s}}{A L^{2}} \times \frac{L^{3} {(\frac{L^{2} ε^{2} - 2 \sin (α_{0}) L L_{f} ε + L_{f}^{2}}{L^{2}})}^{\frac{3}{2}} - L_{f}^{3} + L_{f}^{3} {(\sin (α_{0}))}^{2}}{{(\frac{L^{2} ε^{2} - 2 \sin (α_{0}) L L_{f} ε + L_{f}^{2}}{L^{2}})}^{\frac{3}{2}}}

(8)

Since equation (8) is nonlinear, the stiffness of the device changes constantly with the applied load. Because the proposed device exhibits variable stiffness under a cycle load, its frequency changes constantly, and as a result greatly reduces the possibility of resonance (Humar, 2012).

Equations (7) and (8) were programmed in OpenSees as a new material (see appendix) to represent the behavior of the proposed device (Zheng, 2021). Like equation (4), equation (7) by itself does not manifest snap-through or snap-back behavior, so the coding in OpenSees needs to account for this explicitly. Using the inputted strain (or displacement) values that correspond to Points A, B, E, F in Figure 4 (i.e., the strain at which snap-through occurs, the equilibrium point after snap-through, the strain at which snap-back occurs, and the equilibrium point after snap-back, respectively), the criteria used to determine when snap-through or snap-back occurs are as follows: snap-through occurs when |ε_t | > |ε_t-Δt | and |ε_t | = |ε_A |, whereas snap-back occurs when |ε_t | < |ε_t-Δt | and |ε_t | = |ε_E |, where | | represents absolute value, and ε_t is the strain at the current time step, ε_t-Δt is the strain at the previous time step, ε_A and ε_E are the strains that correspond to Point A and Point E, respectively. Once snap-through or snap-back is detected, the equilibrium point after snap-through or snap-back is used in the next cycle of calculations. In all the numerical analyses performed in the present study, the time step Δt used was the same as that used in the ground motion records.

Other design parameters

As mentioned earlier, helical springs made from steel arranged in the form of an array of von Mises truss are to be used for the proposed device. Helical springs are used because they behave very similar in tension and compression within a certain displacement range. They also possess all the attributes of a good structural device such as high stiffness, high strength, and are relatively easy to analyze. However, for these springs to be able to perform properly, buckling and yielding need to be avoided.

To reduce the possibility of buckling, the following condition should be satisfied (Valsange, 2012)

L_{f} < 4 D

(9)

where L_f is the length and

D

is the coil diameter of the helical spring. If equation (9) cannot be satisfied and buckling becomes an issue, the spring can be placed inside a rigid spring sheath, which will serve as a stabilizer to prevent the spring from bending or buckling.

To avoid yielding, the maximum shear stress in the spring ( $τ_{\max}$ ) should not exceed the shear yield strength ( $S_{s y}$ ) of the material. From Figure 6, the shear stress acting on the cross-section of the wire of a helical spring can be expressed by the equation

τ_{\max} = \frac{16 F}{3 π d^{2}} + \frac{8 F D}{π d^{3}} = k_{s p} \frac{8 F D}{π d^{3}}

(10)

where F is the force acting on the spring, T is the torque, D is the coil diameter of the spring, d is the wire diameter, and

k_{s p} = \frac{2}{3 (\frac{D}{d})} + 1

. To account for the curvature effect of the wire which could increase the stress acting on the inside area of the wire,

k_{s p}

can be modified by the Wahl correction factor (Wahl, 1944) to give

k_{s p} = \frac{4 (\frac{D}{d}) - 1}{4 (\frac{D}{d}) - 4} + \frac{0.615}{(\frac{D}{d})}

Figure 6.

Forces acting on a helical spring.

The shear yield strength of the material ( $S_{s y}$ ) can be estimated from its tensile strength ( $S_{u t}$ ) using the following criterion (Spring Design, 2017)

{{0.35 S}_{u t} \leq S}_{s y} \leq {0.52 S}_{u t}

(11)

Effect of various spring parameters

The effect of several spring parameters that affect the behavior of the proposed device is described in this section. The parameters are the free length of the spring ( $L_{f}$ ), the initial inclination angle ( $α_{0}$ ), the number of spring components ( $N_{s}$ ) and the axial stiffness of the spring ( $k_{s}$ ), which is related to the wire diameter of the spring ( $d$ ), coil diameter of the helical spring ( $D$ ), and number of coils ( $n$ ) by the equation

k_{s} = \frac{G d^{4}}{8 n D^{3}}

(12)

where G is the shear modulus of the material.

The initial stiffness of the device can be obtained by evaluating equation (5) at $u = 0$ , giving

\frac{d P}{d u} = 2 N_{s} k_{s} {(\sin (α_{0}))}^{2}

(13)

which shows the initial stiffness increases as

α_{0}

N_{s}

, or

k_{s}

increases.

The equation for the displacement when snap-through occurs can be obtained by setting equation (5) equal zero and solve for $u$ , giving

u_{s n a p f r o m} = L_{f} (\sin (α_{0}) - \sqrt{{(\cos (α_{0}))}^{\frac{4}{3}} - {(\cos (α_{0}))}^{2}})

(14)

which shows the displacement at which snap-through occurs increases with

L_{f}

and

α_{0}

. Because the snapping distance is approximately equal to

2 L_{f} \sin (α_{0})

, the distance after snap-through occurs also increases with

L_{f}

and

α_{0}

Upon substitution equation (14) into equation (4), the force at which snap-through occurs can be written as

P_{s n a p} = 2 N_{s} k_{s} L_{f} (1 - {(\cos (α_{0}))}^{\frac{2}{3}}) \sqrt{1 - {(\cos (α_{0}))}^{\frac{3}{2}}}

(15)

which shows

P_{s n a p}

increases with an increase in

L_{f}

α_{0}

N_{s}

k_{s}

The effect of these parameters on properties of the device is summarized in Table 2.

Table 2.

Effect of device parameters on device properties.

Parameter	Initial stiffness	Snap force	Snap from	Snap to
Spring Free length ( $L_{f}$ ) $↑$	$-$	$↑$	$↑$	$↑$
Initial inclination angle ( $α_{0}$ ) $↑$	$↑$	$↑$	$↑$	$↑$
Spring components ( $N_{s}$ ) $↑$	$↑$	$↑$	$-$	$-$
Spring axial stiffness ( $k_{s}$ ) $↑$	$↑$	$↑$	$-$	$-$

As for the device’s energy dissipation capacity, it can be seen from Figure 2 that the enclosed area O-A-B-D-E-F-O increases as the snapping force or snapping displacement increases. This is further illustrated in Figure 7 in which the hysteresis loop obtained using the methodology and software discussed earlier for one cycle of loading-unloading-reloading is shown for five values of $L_{f}$ (= 20, 40, 60, 80, 100 mm), with $α_{0}$ = 15, $N_{s}$ = 1, and $k_{s}$ = 8.85 × 10⁵ $N / m$ . As $L_{f}$ increases from $20 m m$ to $100 m m$ , it can be seen in the figure and from Table 3 that it has no effect on the initial stiffness of the device, but both the force at which snap-through occurs and the snap-through displacement from one stable equilibrium point to the next increase. The increase is directly proportional to $L_{f}$ . An increase in $L_{f}$ also increases the size of the loop area, indicating the device is able to dissipate more energy.

Figure 7.

Load-displacement behavior of the proposed device for different spring free Lengths.

Table 3.

Device properties due to change in spring free length.

$L_{f}$ $(m m)$	Initial stiffness ( $N / m$ )	Snap force ( $N$ )	Snap from ( $m m$ )	Snap to ( $m m$ )
20	$1.19 \times 10^{5}$	122	2.2	11.2
40	$1.19 \times 10^{5}$	244	4.4	22.3
60	$1.19 \times 10^{5}$	367	6.7	33.6
80	$1.19 \times 10^{5}$	489	8.9	44.8
100	$1.19 \times 10^{5}$	610	11.1	55.9

The ability of the device to dissipate energy also increases when the initial inclination angle ( $α_{0}$ ), the number of spring components ( $N_{s}$ ), or the spring stiffness ( $k_{s}$ ) increases. From equation (13) and (15), it can be seen that changing $k_{s}$ elicits the same effect as changing $N_{s}$ .

Nonlinear time history analysis of shear frames

Nonlinear time history analyses of structures with energy dissipation systems can generally provide the most realistic indication of global structural responses and demands on individual structural members. Since analytical results from such analyses tend to be more sensitive to small changes in terms of the behavior of structural members and ground motion records used in the analysis, the use of nonlinear response time history analysis with a multitude of ground motions was recommended in the FEMA (1996) guidelines.

In this section, results of nonlinear time history analyses performed on two shear frames to study their seismic performance and to demonstrate the effectiveness of the proposed energy dissipation device are presented. As shown in Figure 8, SF-1 is a one-story shear frame without any energy dissipation device, and SF-1SD is the same shear frame equipped with the proposed device. The height $h$ of the frames is $3.6 m$ and the bay width $L$ is $6.0 m$ . The beam was assumed to be very stiff when compared to the columns and so both frames can be modeled as single degree-of-freedom systems. The properties of the columns and supporting members are given in Table 4.

Figure 8.

One-story shear frames (SFs).

Table 4.

Shear frame properties.

Columns	$W 6 \times 15 (W 150 \times 22.5)$
Beam	Infinitely rigid
Braces	$H S S 5 \times 5 \times 1 / 4 (H S S 127 \times 127 \times 6)$

The proposed device installed in SF-1SD is composed of helical springs with a free length of 25.4 $m m$ , a combined initial spring stiffness of $90 k N / m m$ , and an initial inclination angle of $15 °$ . Using these parameters, the external force at snap-through was determined to be $15.52 k N$ and the amount of snap displacement was determined to be $14.2 m m$ . The amount of energy that this device can dissipate in one cycle of snap-through and snap-back is approximately 441 J.

Peak story drift (PSD) and residual story drift (RSD) are used as the main engineering performance indices to evaluate the seismic behavior of these shear frames. PSD and RSD are often used to quantify the seismic behavior of structures because the former gives information on the amount of elastic and inelastic displacements experienced by the structure during an earthquake, while the latter provides insight into the amount of inelastic displacement remained in the structure after the ground excitation ends. As shown in Table 1, the performance level of a building can be determined based on its PSD and RSD values.

Both frames were assumed to be an office building with a seismic weight (ASCE/SEI 7–22, 2022) of

3564 k g

and

3727 k g

, respectively. The initial natural periods and natural frequencies

ω_{n}

were calculated and are shown in Table 5. The addition of the device, which possesses its own mass, damping and stiffness characteristics, changes the dynamic properties of the frame since all three system parameters (mass, damping and stiffness) that affect the response of a structure under a dynamic excitation have been changed.

Table 5.

Dynamic properties of the shear frames.

Frame	Initial lateral stiffness ( $10^{6} N / m$ )	Initial natural period ( $s$ )	$ω_{n}$ ( $r a d / s$ )
SF-1	1.255	0.335	18.76
SF-1SD.	13.105	0.104	60.42

The damping coefficient $c$ was calculated using the equation $c = 2 ζ m ω_{n}$ , where $ζ$ is the damping ratio, (taken as 5%), $m$ is the seismic weight, and $ω_{n}$ is the natural frequency of the frames.

These frames were assumed to be located in downtown Los Angeles area on stiff soil of site class D. The ground motion histories used in the nonlinear dynamic analyses were developed in the SAC Joint Venture Steel Project Phase two (Somerville, 1997). The earthquake suites corresponding to downtown Los Angeles were selected for seismic hazard levels that correspond to a 50% (frequent), 10% (rare) and 2% (very rare) probability of exceedance in 50 years. Each suite contains 20 ground motion records. These 60 records were derived from fault-parallel and fault-normal orientations of 30 earthquake records with adjustment in amplitude and frequency domain. Some ground motion characteristics for these earthquakes are summarized in Table 6.

Table 6.

Characteristics of ground motion records.

Records		Magnitude	Duration ( $s$ )	Peak ground acceleration ( $m / s^{2}$ )
LA01-20	Rare	6.0–7.3	14.945–79.98	2.30–9.99
LA21-40	Very rare	6.7–7.4	14.945–59.98	4.10–13.04
LA41-60	Frequent	5.7–7.7	26.14–79.98	1.09–7.75

The structural models of the two shear frames were constructed using the TCL file to run in OpenSees. The new material added and implemented into the framework was assigned to a member to simulate the snap-through and snap-back behaviors of the proposed device.

The story lateral displacements of the two shear frames under a specific earthquake (LA28) as shown in Figure 9 are plotted in Figure 10. This LA28 ground motion record has a peak acceleration of $1.331 g$ . From Figure 10, it can be seen that the shear frame without the device has a peak story displacement of $152.27 m m$ and a residual displacement of $99.01 m m$ . These numbers are the same as those reported by Tang and Lui (2014). When compared with SF-1, SF-1SD only exhibits a peak story displacement of $15.76 m m$ , and no residual story displacement. Thus, the frame equipped with the proposed device behaves much better under this earthquake.

Figure 9.

Ground motion record – LA28 (1994 northridge earthquake).

Figure 10.

Story displacement history of the two frames under LA28.

Nonlinear time history analyses were then performed on the two frames using all 60 earthquake records (LA01 to LA60). The results from these analyses are summarized and shown in Figure 11 in which the PSDs and RSDs are plotted for the three levels of earthquake severity. The vertical lines in the figures show average values of the corresponding PSDs and RSDs. It can be seen from these plots that as the earthquake severity level gets higher, both peak and RSDs increase. At the same severity level, SF-1SD shows noticeable smaller peak and RSDs, indicating the benefits of adding the proposed device.

Figure 11.

Peak and residual story drifts of two shear frames under three Levels of earthquakes.

Under frequent, rare, and very rare earthquakes, the frame without the proposed device (SF-1) gives average PSDs of $31.85 m m$ , $43.40 m m$ and $69.35 m m$ , and peak story residual drifts of $2.39 m m$ , $5.44 m m$ , and $17.64 m m$ , respectively. When compared to SF-1, the frame with the device (SF-1SD) performs much better in terms of reducing both the peak and RSDs. The average PSDs for SF-1SD under the three severity levels of earthquakes are $9.84 m m$ , $11.16 m m$ , and $14.77 m m$ . They are all below the recommended transient drift ratio of 0.5% for braced steel frames under the performance level of Immediate Occupancy (Table 1).

Snap-through and snap-back behaviors are observed for all three levels of earthquake severity, especially under very rare earthquakes. The proposed device also provides some initial stiffness to the frame to help control lateral displacements before snap-through occurs. These results show that the proposed device can control PSDs as well as dissipating energy.

Nonlinear time history analysis of multistory steel frames

The frames used for this study are shown in Figure 12. They consist of a four-story moment resisting frame (MRF-4), a Chevron braced frames (CBF-4) and a moment frame equipped with the proposed energy dissipation device (CBF-4SD). All frames have the same dimensions with a story height of $3.6 m$ and a bay width of $8.0 m$ . The beams are rigidly connected to the columns. The sections and section properties used for these frames are given in Table 7 (Steel Construction Manual, 2017). All member cross-sections are oriented in such a way that bending about the strong axis would occur in the plane of the frame. The yield strength of all beams and columns is $248 M P a$ and that of the bracing members is $345 M P a$ . All members have a modulus of elasticity of $200 G P a$ .

Figure 12.

Structural models of the three four-story frames.

Table 7.

Properties of frame members.

Member	Frame member	Cross section area ( $m^{2}$ )	Moment of inertia ( $m^{4}$ )	Plastic section modulus ( $m^{3}$ )
Column	$W 310 \times 202$ $(W 12 \times 136)$	$2.57 \times 10^{- 2}$	$5.16 \times 10^{- 4}$	$3.51 \times 10^{- 3}$
Beam	$W 310 \times 158$ $(W 12 \times 106)$	$2.01 \times 10^{- 2}$	$3.88 \times 10^{- 4}$	$2.69 \times 10^{- 3}$
Bracing member	$H S S 254 \times 152 \times 8$ $(H S S 10 \times 6 \times 5 / 16)$	$5.65 \times 10^{- 3}$	$2.22 \times 10^{- 5}$	$2.92 \times 10^{- 4}$

Of the three frames, MRF-4 is the most flexible and the lightest while CBF-4 is the stiffest and is heavier. The stiffness of CBF-4SD falls between that of MRF-4 and CBF-4 and weighs more or less the same as CBF-4. Under the same ground excitation, it is anticipated that MRF-4 will experience the largest drifts and CBF-4 the smallest, with CBF-4SD somewhere in between. In terms of energy dissipation, MRF-4 dissipates energy exclusively through yielding or inelastic deformations of its beams and columns; CBF-4 dissipates energy through inelastic deformations of its beams and columns as well as inelastic buckling of the braces; and CBF-4SD dissipates energy primarily by snap-through and snap-back actions of the proposed device.

All structural members (beam, columns, and bracing members) were modeled using force-based (FB) fiber beam-column element in OpenSees. The bracing member was modeled using Yang’s method (Yang et al., 2009), which allowed for inelastic bracing member behavior such as yielding in tension and buckling in compression to be accounted for (Karmazinva and Melcher, 2014; Padilla-Llano et al., 2015).

The proposed energy dissipation device used in all the nonlinear time history analyses in this section is composed of helical springs with a free length of 55 $m m$ , a combined initial stiffness of $810 k N / m m$ , and an initial inclination angle of $15 °$ . The external force that would cause snap-through was $302.4 k N$ and the snap-through displacement was $30.77 m m$ . The displacement of the device after snap-through occurs is well below the 1.5% transient drift limit under the performance level of Life Safety for braced steel frame (Table 1). The amount of energy that this device can dissipate in one cycle of snap-through and snap-back is approximately 18.6 kJ.

In dynamic analysis of multistory frames, the lateral force acting on each story comes from the lateral acceleration of the ground motion acting on the effective seismic weight of that story. Assuming all the frames are office buildings, the seismic weights would include the dead load (DL) plus a partition weight (PW) (AISC/SEI 7-22, 2022). These effective seismic weights are summarized in Table 8. They were modeled as lumped masses placed at mid-span of the beams.

Table 8.

Effective seismic weights of the four-story frames.

Story	DL (kg)		PW (kg)	Total (kg)
Story	MRF	CBF	PW (kg)	MRF	CBF
4	28,087	28,342	0	28,087	28,342
3	28,814	29,324	4,697	33,511	34,021
2	28,814	29,324	4,697	33,511	34,021
1	28,814	29,324	4,697	33,511	34,021

In addition to the lateral load induced by the ground motion, there are vertical DL and live load (LL) acting on the frame. The DL was calculated by multiplying the dead weight portion of the seismic weight by

9.81 m / s^{2}

and applied uniformly on each floor. A live to DL ratio of

2

was used to calculate the live load. The total vertical load was converted to an equivalent uniformly distributed load

w_{e q}

acting on the beam of each floor as shown in Table 9. The corresponding equivalent axial force

P_{e q}

and beam end moments

M_{F}

were applied to the beam-column joint in the frame model shown in Figure 13. This method of modeling is consistent with what is being used in OpenSees.

Table 9.

Equivalent distributed loads.

Story	DL (kN)		LL (kN)	Total (kN)		$w_{e q}$ (kN/m)
Story	MRF	CBF	LL (kN)	MRF	CBF	MRF	CBF
4	276	278	552	828	830	103	104
3	283	288	566	849	854	106	107
2	283	288	566	849	854	106	107
1	283	288	566	849	854	106	107

Figure 13.

Load pattern conversion used in opensees.

Using an eigenvalue analysis (Chopra, 2017; Cruz and Miranda, 2017), the initial fundamental frequencies and periods (

ω_{1}

and

T_{1}

) of the frames were calculated and the results are summarized in Table 10.

Table 10.

Initial fundamental frequencies and periods.

	$ω_{1}$ $(r a d / s)$	$T_{1}$ $(s)$
MRF-4	6.71	0.937
CBF-4SD.	17.36	0.362
CBF-4	27.08	0.232

The same earthquake suites (Sommerville, 1997) used in the analysis of the shear frames were used here. In addition to obtaining the peak story displacements (PSDs) and residual story displacements (RSDs) of the top story of the frames, peak inter-story drift ratio (IDR) and residual IDR were also obtained and used as performance indices to evaluate the effectiveness of the proposed device. Note that ASCE/SEI 7–22 (2022) and FEMA 273 (1996) place restrictions on how large the peak and/or residual IDR a structure can experience. They are therefore important performance indices for evaluating earthquake damage and are often used in PBSD.

Results obtained for the peak top story displacements and peak inter-story drift ratios of these three frames subject to the three earthquake excitation levels are shown in Figures 14 and 15, respectively.

Figure 14.

Peak top story displacements under frequent, rare, and very rare earthquakes.

Figure 15.

Peak inter-story drift ratios under frequent, rare, and very rare earthquakes.

It can be seen that the average peak top story displacement of MRF-4 is larger than the other two frames, and the displacements increase as the severity level of the seismic excitation increases. The results are as expected given that the lateral stiffness of MRF-4 is the lowest. When comparing CBF-4SD and CBF-4, CBF-4 has a smaller peak top story displacement. This is due to its larger lateral stiffness. However, some bracing members of CBF-4 buckled under rare and very rare earthquakes. The use of the proposed device in CBF-4SD can help reduce top story displacement by providing not only stiffness to, but allowing energy to be dissipated from, the frame. Under frequent earthquakes, all three frames show acceptable peak inter-story drift ratios under life safety (Table 1). However, for rare and very rare earthquakes, only CBF-4 and CBF-4SDs (except for the 1^st story under very rare earthquakes) satisfy the inter-story drift criterion. Under very rare earthquakes, the lower stories of MRF-4 show rather large peak inter-story drifts, and this could lead to noticeable structural or non-structural damage.

In addition to the PSDs discussed above, residual top story displacements and inter-story drift ratios were also obtained for the three levels of seismic excitation. The results are plotted in Figures 16 and 17.

Figure 16.

Residual top story displacements under frequent, rare, and very rare earthquakes.

Figure 17.

Residual inter-story drift ratios under frequent, rare, and very rare earthquakes.

As the earthquake severity level increases, the residual top story displacement of each frame type also increases. Among them, MRF-4 has the largest mean residual top story displacement due to permanently inelastic deformation and insufficient damping. Under frequent earthquakes, both CBF-4 and CBF-4SD exhibit noticeably smaller residual top story displacements. When compared to CBF-4 under rare and very rare earthquakes, the residual top story displacements for CBF-4SD are larger, but they are still considered small when compared to those of MRF-4. In terms of residual top story displacements, CBF-4 performs the best. The frame remains mostly elastic, but under very rare earthquakes residual top story displacement was observed when the brace members buckled. As for the inter-story drift ratio, all three frames perform satisfactorily under frequent earthquakes. The residual inter-story drift ratios for CBF-4 and CBF-4SD are almost negligible. As the severity level of earthquake excitations increases, the inter-story drift ratios increase for all three frames but are more pronounced for MRF-4. Under rare and very rare earthquakes, the residual inter-story displacements for CBF-4 and CBF-4SD are considered satisfactory, although those for CBF-4SD are larger given that CBF-4 has a larger lateral stiffness. However, when compared to MRF-4, noticeable reduction in the residual inter-story displacements is observed. Thus, the effectiveness of the proposed device is demonstrated.

The maximum bracing member compression force under three levels of seismic severity are shown in Figure 18. For ease of comparison, the same scale was used to plot the figures. Since there are no bracing members in MRF-4, their values are shown as zero in Figure 18.

Figure 18.

Maximum bracing member compressive force under frequent, rare and Very rare earthquakes.

From the figure, it can be seen that an increase in the seismic severity level will result in an increase in the maximum bracing member compression force. For all three seismic severity levels, the bracing forces in CBF-4SD are smaller than those in CBF-4. Given that the inelastic buckling capacity of the bracing member used in CBF-4 and CBF-4SD is 1138 kN (Steel Construction Manual, 2017), it can be seen that a number of bracing members in CBF-4 experience buckling under rare or very rare earthquakes, but a large percentage of bracing members in CBF-4SD continue to remain intact.

Summary and conclusions

In this research, a novel energy dissipation device that makes use of the snap-through and snap-back behaviors of springs was proposed. The proposed device is composed of an array of helical steel springs connecting an outer tube and an inner tube at a specific inclination angle. A rigid bar with stationary bushings is connected to the beam member and it can slide inside the inner tube. These bushings, when in contact with one end of the inner tube, can push the inner tube and apply load to the spring, thereby causing snap-through to occur in the spring and dissipate energy. The energy dissipation characteristics of the device are affected by a number of parameters. The effects of these parameters on the behavior of the proposed device were studied. The constitutive equation that describes the behavior of this device was derived and programmed as a new material in the open-source software OpenSees. The software was then used to perform nonlinear time history analyses to demonstrate the effectiveness of the device in controlling peak and residual deformations of shear and moment resisting frames. Some general conclusions from the research are summarized below.

(1) The behavior of the proposed energy dissipation device is related to the wire diameter ( $d$ ), coil diameter ( $D$ ), number of coils ( $n$ ), free length ( $L_{f}$ ) and initial inclination angle ( $α_{0}$ ) of the spring, as well as the number of spring components ( $N_{s}$ ) used. The snap-through force (i.e., the force at which snap-through occurs) and the snap-to displacement (i.e., the rest position of the spring component after snap-through) can be adjusted by increasing or decreasing the corresponding parameters based on design requirements. The snap-through force can be increased by increasing $d, L_{f}, α_{0}$ and $N_{s}$ , or decreasing $n$ and $D$ . The snap-to distance can be increased by increasing $L_{f}$ and $α_{0}$ . The initial stiffness of the device can be increased by increasing $α_{0}$ , $N_{s}$ , and the axial stiffness of the spring $k_{s}$ .

(2) To simplify the structural modeling, the load-deflection behavior of the proposed device was modeled as a new material, programmed in C++ and implemented in OpenSees. The added material can be directly assigned to any member types in the element library of OpenSees. By using this new material, the behavior of the device can be readily incorporated in OpenSees, making the modeling less time-consuming and the analysis more robust. The input needed for this command is related to the spring parameters. This simulation process was verified, and it was demonstrated that this newly defined material could be used to simulate the snap-through and snap-back behaviors of the proposed device.

(3) A series of nonlinear time history analyses was performed on a one-story shear frame with and without the proposed device. It was shown that the frame equipped with the device exhibited better performance in terms of reducing peak and residual story displacements when compared to the shear frame without the device. Even under very rare earthquakes, the device-equipped frame showed satisfactory results. The residual story displacement was small.

(4) Nonlinear time history analyses were then performed on three four-story steel frames - a moment resisting frames without the device, a Chevron braced frame, and a moment resisting frame with the device - under three different levels of seismic severity. The results showed that the Chevron braced frame experienced the least peak and residual top story displacements as well as the least peak and residual inter-story displacements. However, a number of bracing members with high axial force underwent buckling. When compared with the moment resisting frame, the frame equipped with the proposed device was shown to experience noticeable smaller peak and residual top and inter-story displacements, especially under rare and very rare earthquakes. When compared to the Chevron braced frame, the frame equipped with the proposed device was shown to experience slightly higher peak and residual top and inter-story displacements, but no buckling was observed for the majority of the supporting members. Though larger than those of the Chevron braced frame, all peak and residual displacements of the device-equipped frame still satisfy the PBSD requirements for life safety.

Footnotes

Acknowledgment

The first author, recipient of the Wen-Hsiung and Kuan-Ming Li Fellowship, would like to thank the sponsor of this fellowship for providing him with financial support to conduct this research.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Eric M Lui

Appendix

The codes used in the TCL file to invoke the definition of the material in OpenSees are: uniaxialMaterial TCSpringDevice $MaterialTag $epsA $epsB $epsE $epsF $E0 $AA $BB $Alpha0

where

uniaxialMaterial is the OpenSees command for a uniaxial material.

TCSpringDevice is the name of the device.

$MaterialTag is the user defined tag of the device.

$epsA , $epsB , $epsE , and $epsF are the strain (or displacement) values that correspond to points A, B, E, and F in Figure 4.

$E0 is the initial pseudo modulus of the device.

$AA and $BB are parameters introduced to simplify the programming codes, in which $$ A A = N_{s} K_{s} L$ and $$ B B = \frac{L_{f}}{L}$ .

$Alpha0 is the initial inclination angle of the spring component.

Once the parameters and the configurations of the helical springs are determined, the values for all of the above input parameters can be obtained, and the stress-strain (or load-displacement) behavior of the material used to represent the device are now defined.

References

ASCE/SEI 7-22 (2022) Minimum Design Loads and Associated Criteria for Buildings and Other Structures. Reston, VA: American Society of Civil Engineers.

Avramov

Mikhlin

(2004) Snap-through truss as a vibration absorber. Journal of Vibration and Control 10: 291–308.

Chopra

(2017) Dynamics of Structures – Theory and Applications to Earthquake Engineering. 5th edition. Hoboken, NJ: Pearson.

Connor

(2003) Introduction to Structural Motion Control. Englewood Cliff, NJ: Prentice Hall.

Cruz

Miranda

A critical review of the Rayleigh damping model. Paper No. 2674. In” 16th World Conference on Earthquake 16WCEE, Santiago Chile, January 9–13 2017. https://www.wcee.nicee.org/wcee/article/16WCEE/WCEE2017-2674.pdf

Deng

Cheng

Liang

, et al. (2020) Topology optimization for energy dissipation design of lattice structures through snap-through behavior. Computer Methods in Applied Mechanics and Engineering 358: 112641. DOI: 10.1016/j.cma.2019.112641

FEMA-273 (1996) NEHRP Guidelines for the Seismic Rehabilitation Buildings. Washington D.C.: Federal Emergency Management Agency.

FEMA-356 (2000) Prestandard and Commentary for the Seismic Rehabilitation of Buildings. Washington D.C: Federal Emergency Management Agency.

Lakes

Plesha

(2018) Design, fabrication, and analysis of lattice exhibiting energy absorption via snap-through behavior. Materials and Design 141: 426–437. DOI: 10.1016/j.matdes.2017.12.050

10.

Hrinda

(2010) Snap-through instability patterns in truss structures. In: 51st AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference, structures, structural dynamics, and materials and co-located conferences, Orlando, April 12-15, 2010.

11.

Huang

Vahidi

(1971) Snap-through buckling of two simple structures. International Journal of Non-Linear Mechanics 6(3): 295–310. DOI: 10.1016/0020-7462(71)90011-4

12.

Humar

(2012) Dynamics of Structures. London, UK: CRC Press, Taylor & Francis Group.

13.

Kala

Kalina

(2016) Static equilibrium states of von misses trusses. International Journal of Mechanics 10: 294–298.

14.

Karmazinova

Melcher

(2014) Initial imperfections of steel and steel-concrete composite columns subjected to buckling compression. WSEAS Transactions on Applied and Theoretical Mechanics 9: 27–34.

15.

Koiter

(1945) On the stability of elastic equilibrium. Doctoral Dissertation: Delft Polytechnic Institute.

16.

Fang

Duan

(2020) High structural damping based on the internal snap-buckling mechanism of a continuous metal module. Mechanics Research Communications 105: 103514. DOI: 10.1016/j.mechrescom.2020.103514

, et al. (2013) Dynamic response of a chevron concentrically braced frame. Journal of Structural Engineering 139(4): 515–525. DOI: 10.1061/(ASCE)ST.1943-541X

18.

Ozbulut

Hurlebaus

(2011) Re-centering variable friction device for vibration control of structures subjected to near-field earthquakes. Mechanical Systems and Signal Processing 25(8): 2849–2862. DOI: 10.1016/j.ymssp.2011.04.017

(2015 March 24-27) OpenSees simulation of steel column axial cyclic response including local buckling. In: Proceedings of the Annual Stability Conference. Nashville, TN: Structural Stability Research Council. https://www.aisc.org/globalassets/continuing-education/ssrc-proceedings/2015/opensees-simulation-of-steel-column-axial-cyclic-response-including-local-buckling.pdf

20.

Pecknold

Ghaboussi

Healey

(1985) Snap-through and bifurcation in a simple structure. Journal of Engineering Mechanics 111(7): 909–922.

21.

Psotny

Ravinger

(2003) Von misses truss with imperfection. Slovak Journal of Civil Engineering 2003(2): 1–7. https://www.svf.stuba.sk/buxus/docs/sjce/2003/2003_2/file5.pdf

22.

Qammer

Dalal

(2016) Development of performance based seismic design: a review. In: Proceedings, International Conference on Engineering and Technology (ICET), Beijing, China, June 14-15, 2016, 5pp.

23.

Sano

Wada

(2018) Snap-buckling in asymmetrically constrained elastic strips. Physical Review E 97(1): 013002. https://journals.aps.org/pre/abstract/10.1103/PhysRevE.97.013002

24.

Savi

Nogueira

(2010) Nonlinear dynamics and chaos in a pseudoelastic two-bar truss. Smart Materials and Structures 19: 115022, 11pp.

25.

Schenk

Guest

(2013) On zero stiffness. Journal of Mechanical Engineering Science 228(10): 1701–1714. DOI: 10.1177/0954406213511903

26.

Snirc

Ravinger

(2017) Statics and dynamics of snap-through effect. Procedia Engineering 190: 540–546. DOI: 10.1016/j.proeng.2017.05.377

27.

Somerville

(1997) Development of Ground Motion Time Histories for Phase 2 of the FEMA/SAC Steel Project. Sacramento, CA: SAC Joint Venture.

28.

Sonnerlind

(2014) Buckling, when structures suddenly collapse. Available at: https://www.comsol.com/blogs/buckling-structures-suddenly-collapse

29.

Spring Design (2017). Available at: https://www.tokaibane.com/en/spring-design/compression-springs-formulas

30.

Steel Construction Manual (2017). 15th edition. American Institute of Steel Construction.

31.

Tang

Lui

(2014) Hybrid recentering energy dissipative device for seismic protection. Journal of Structures 262409: 1–17. DOI: 10.1155/2014/262409

32.

Tarnai

(2003) Zero stiffness elastic structures. International Journal of Mechanical Sciences 45(3): 425–431. DOI: 10.1016/S0020-7403(03)00063-8

(1991) Large-rotation snap-through buckling in laminated cylindrical panels. Finite Elements in Analysis and Design 9(1): 65–75. DOI: 10.1016/0168-874X(91)90019-U)

34.

Valsange

(2012) Design of helical coil compression spring - a review. International Journal of Engineering Research in Africa 2(6): 513–522. http://www.ijera.com/papers/Vol2_issue6/BY26513522.pdf

35.

Wahl

(1944) Mechanical Springs. Cleveland, OH, USA: Penton Publishing Company.

36.

Wiebe

Virgin

Stanciulescu

, et al. (2011) On snap-through buckling. In: 52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials.

37.

Yang

Moehle

Stojadinovic

(2009) Performance Evaluation of Innovative Steel Braced Frames, Pacific Earthquake Engineering Research Center. https://peer.berkeley.edu/sites/default/files/web_peer9103_t._y._yang_jack_p._moehle_and_bozidar_stojadinovic.pdf

38.

Zhang

Tichem

van Keulen

(2021) Rotational snap-through behavior of multi-stable beam-type metastructures. International Journal of Mechanical Sciences 193: 106172. DOI: 10.1016/j.ijmecsci.2020.106172

, et al. (2020) Tunable energy trapping through contact-induced snap-through buckling in strips with programmable imperfections. Extreme Mechanics Letters 37: 100732. DOI: 10.1016/j.eml.2020.100732

, et al. (2020) Deterministic snap-through buckling and energy trapping in axially-loaded notched strips for compliant building blocks. Smart Materials and Structures 29: 02LT03. DOI: 10.1088/1361-665X/ab6486

41.