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Fundamentally, BRBs are a mild steel plate (core) confined by a relatively stronger case (casing) to prevent the plate from buckling in compression. Fig. 1-a shows a schematic of a typical BRB. The steel core is either a cruciform or a plate and the casing is generally a steel tube filled with concrete. A really small gap is left between the core plate and the confining concrete, big enough to allow the steel core to initiate buckling but not so large to let it buckle. This gap also accommodates the change in volume of the core steel while in compression. This entire mechanism lets BRBs exhibit a balanced tension-compression behavior, opposite to a simple steel beam brace. (Fig. 1-b)

**BRB****.tcl**(Or go to**GitHub**) and**SolverAlgorithms.tcl****Modeling of Buckling Restrained Brace (BRB)****A Quick Intro to BRBs**Fundamentally, BRBs are a mild steel plate (core) confined by a relatively stronger case (casing) to prevent the plate from buckling in compression. Fig. 1-a shows a schematic of a typical BRB. The steel core is either a cruciform or a plate and the casing is generally a steel tube filled with concrete. A really small gap is left between the core plate and the confining concrete, big enough to allow the steel core to initiate buckling but not so large to let it buckle. This gap also accommodates the change in volume of the core steel while in compression. This entire mechanism lets BRBs exhibit a balanced tension-compression behavior, opposite to a simple steel beam brace. (Fig. 1-b)

**Fig. 1**A schematic of a typical BRB (left,

*Upadhyay & Pantelides, 2019*) and the concept (right)

**What are we aiming for?**

Comparing the tension and compression backbones of a BRB hysteresis (refer to the output at the end of this post), we can see that BRBs exhibit higher capacity in compression as compared to tension. In tension, the core place is elongating and yielding similar to a coupon test. But in compression, once the core plate start to buckle, it touches and rubs (Fig. 1) against the internal surface of the concrete (or the casing). The friction between the two surfaces adds to the compression capacity of a BRB.

Our OpenSees model should aim for this behavior. Here, we’ll use

*Steel02*material model for the steel core. Since

*Steel02*is a symmetric in tension and compression, we’ll add

*Pinching4*material in parallel to

*Steel02*cause the unbalance in BRB output.

**OpenSees Model**

In this OpenSees model we will try to simulate a cyclic pseudo static component test of a BRB using a 2-dimensional model. As usual, let’s first define our basic units (inch, kip and sec), some dependent units and constants.

Now we’ll define the geometry of the BRB (shown in Fig 2):

L_WP: Work point length, length of the BRB element from node-to-node. (See fig. 3)

L_Core: Length of the BRB core

LR_BRB: Length Ratio, the ratio of core length to work point length

A_Core : Cross-sectional area of the core plate

A_End : Cross-sectional area of the end section of the core.

L_WP: Work point length, length of the BRB element from node-to-node. (See fig. 3)

L_Core: Length of the BRB core

LR_BRB: Length Ratio, the ratio of core length to work point length

A_Core : Cross-sectional area of the core plate

A_End : Cross-sectional area of the end section of the core.

**Fig. 2**Various parts of a BRB core and their stiffness (top); definition of work-point length (bottom)

The two end parts and the core act as springs in series combination. The axial stiffness of each part of the BRB is shown in Fig. 2 above. To simplify our BRB model, we would want to replace this series combination of three springs with a single spring with equivalent stiffness using the following equations,

*E*

*wp*is the equivalent modulus of elasticity of the brace if it were to be replaced with a steel member from workpoint-to-workpoint. Now, for convenience, we can assume that the end section area is much larger than the core sectional area. This simplifies the above equation to become,

We’ll use this equivalent elasticity modulus to model our BRB. Now, we will define four materials as follows,

BRBMaterial_1:

*Steel02*material has a symmetric hysteresis with isotropic and kinematic hardening properties, with no failure point.

BRBMaterial_2:

*Pinching4*material is used to add additional strength to the BRB material in compression due to friction.

BRBMaterial_3: A new material is created by combining

*Steel02*and

*Pinching4*materials in

*parallel*combination.

BRBMaterial_4:

*Fatigue*material is then used to wrap

*BRBMaterial_3*to simulate failure due to cyclic fatigue and/or ultimate tensile/compression strain.

Now define

*Pinching4*material model. The parameters used here (shown below,**Table 1**) are validated using the BRB component tests done at the University of Utah and are published in**Upadhyay and Pantelides (2019)**. The purpose of this pinching material is to add compression strength to the BRB to achieve the unbalance in the hysteresis.The inputs are values of stress (ksi) at corresponding strain (in./in.) values, defined as Tcl lists. The stress addition in positive (tension) direction is small but in compression (negative).

Now, combine the

*Steel02*and*Pinching4*materials in parallel combination to define a new material with tag*BRBMaterial_3*. This*BRBMaterial_3*is then wrapped by the Fatigue material to define*BRBMaterial_4*that includes failure due to peak strain and cyclic fatigue. The parameters for the*Fatigue*material are also provided in**Table 1**of*Upadhyay and Pantelides (2019)*. Based on several published test data, the BRB core is considered to achieve a maximum of 3.5% strain.Now that we have defined our materials, we proceed to define nodes at the work points. Define boundary conditions using Single Point Constraints. Fix the bottom node in all three DOFs and fix the top node in rotation DOF only.

**Fig. 3**OpenSees model for the BRB component test.

Corotational truss element is a good choice to simulate any brace. Truss elements do not transfer any moment at the connections. This corotational truss element is defined from workpoint-to-workpoint and has cross-sectional area of the BRB core and

*BRBMaterial_4*.Now, let’s go ahead to perform a cyclic analysis using a displacement-controlled integrator. The test protocol (displacement protocol) is shown below. Before performing analysis, we will define recorders to record BRB axial force, axial deformation and cyclic fatigue in a specific output folder "BRB_Cyclic_Test"

**Fig. 4**Test protocol.

And, here is the output. We can compare the hysteresis (Fig. 5 a) of out BRB model with the one tested (Test 7,

*Xu & Pantelides 2017*) and plot the cyclic fatigue damage index (Fig. 5 b) in the core. Once the damage index reaches 1.0, the material fails. This failure is also shown on the hysteresis.

**Fig. 5**(a) Comparison of cyclic hysteresis; (b) damage index of the core steel.

**References:**1) Upadhyay and Pantelides (2019), "Residual drift mitigation for bridges retrofitted with buckling restrained braces or self centering energy dissipation devices." Eng. Struct,10.1016/j.engstruct.2019.109663

2) Xu W, Pantelides CP., (2017). "Strong-axis and weak-axis buckling and local bulging of buckling-restrained braces with prismatic core plates."

*Eng Struct*2017;153:279–89.

3) McKenna, F., Scott, M. H., and Fenves, G. L. (2010) “Nonlinear finite-element analysis software architecture using object composition.”

*Journal of Computing in Civil Engineering*, 24(1):95-107.