E.8 Interface Modeling using Spring and Interface Elements

E.8.1 Spring and Interface Element

As shown in the figure, the responses of springs (Spring, EarthSpring) and interface elements (IL22, ILL33) under pure shear conditions were verified.

Figure E.8.1 Verifcation Models of Spring and Interface Elements

When the calculation is performed under displacement control up to \(\small \Delta = 0.5\), the responses should be identical in all cases, as shown in the table.

Table E.8.1 Analysis Results

Time	Displacement	Force(IAX4)*	Force(Other Elements)*	strain	stress
0.1	0.05	0.7854	0.25	0.05	0.25
0.2	0.1	1.5708	0.5	0.1	0.5
0.3	0.15	2.3562	0.75	0.15	0.75
0.4	0.2	3.1416	1	0.2	1
0.5	0.25	2.87983	0.91667	0.25	0.91667
0.6	0.3	2.61796	0.83333	0.3	0.83333
0.7	0.35	2.3562	0.75	0.35	0.75
0.8	0.4	2.09443	0.66667	0.4	0.66667
0.9	0.45	1.83257	0.58333	0.45	0.58333
1	0.5	1.5708	0.5	0.5	0.5

* Sum of nodal force when Interface element

Input file

V-Spring.inp : Spring
V-EarthSpring.inp : EarthSpring
V-I2D4.inp : I2D4 (2+2 2-dimensional Interface Element) using uncoupled material
V-IAX4.inp : IAX4 (2+2 Axysymmetric Interface Element) using uncoupled material
V-I3D6.inp : I3D6 (3+3 3-dimensioal Interface Element) using uncoupled material
V-I3D8.inp : I3D8 (4+4 3-dimensioal Interface Element) using uncoupled material
V-I2D4-C.inp : I2D4 (2+2 2-dimensional Interface Element) using cohesive material
V-IAX4-C.inp : IAX4 (2+2 Axysymmetric Interface Element) using cohesive material
V-I3D6-C.inp : I3D6 (3+3 3-dimensioal Interface Element) using cohesive material
V-I3D8-C.inp : I3D8 (4+4 3-dimensioal Interface Element) using cohesive material

E.8.2 Adhesive Joint

The second example focuses on the analytical solutions for adhesive joints, which are often used as fundamental tests in the field of structural repair and strengthening. As shown in the figure, adhesive joints can be classified as pull–push or pull–pull joints depending on the boundary conditions. Yuan et al. (2001) and Wu et al. (2002) derived four analytical solutions for idealized pull–push and pull–pull joints under the following assumptions: (1) the adherends are homogeneous and linearly elastic, (2) bending of the adherends is neglected, with normal stresses on the cross-section assumed to be uniformly distributed, and (3) the adhesive layer is subjected only to shear forces, with constant thickness and width. These solutions cover cases where the bond–slip relationship in the adhesive layer is either linear or bilinear.

Figure E.8.2 Types of Adhesive joints

Figure E.8.3 Analytic solution of adhesive joints (Wu et al. ,2002)

Figure E.8.4 Analytic solution of pull-push adhesive joints with Model II(Wu et al. ,2002)

In this verification example, as shown in Figure E.10.4, the pull–push condition was applied to a 10 cm × 10 cm adhesive joint (bond width: 4 cm, bond length: 35 cm). The interface was modeled using spring elements (Spring) and interface elements (IL22), and the results were compared with the analytical solution. The CFRP plates placed at the top and bottom, as well as the concrete, were modeled using truss elements. Figure xx presents the analytical solution for a pull–push adhesive joint with a bilinear bond–slip model.

Figure E.8.5 Analysis Model

Table E.8.2 Properties of Analysis Model (Adhesive length 35cm)

Classification

Material Properties

Remark

CFRP plate

E₁=2.35E6 kgf/cm²

b₁=4cm , t₁=0.0167cm

T3D2 Element

Concrete

( Strength 270kgf/cm² )

E₂=246,475kgf/cm²

b₂=10cm, t₂=10cm

T3D2 Element

Bon-slip model(Model II)

δ₁=0.02cm, δ_f= 0.08cm,

τ_f = 40kgf/cm²

Spring or I2D4 Element

In the analytical solution, the ultimate load was calculated as 1406.4 kgf. Figure E.10.6(a) compares the load–displacement curves from the analytical solution, the spring element model, and the interface element model. Figures E.10.6(b)–E.10.6(d) show the distribution of shear bond stress along the interface with increasing load for both the analytical and numerical solutions. Figure E.10.7 compares the shear bond stress distribution from each numerical solution with that from the analytical solution. A slight time delay effect was observed in load transfer. In the case of the spring element model, convergence was not achieved at a maximum load of 1400 kgf/cm², so the analysis was performed up to 1300 kgf/cm².

Figure E.8.6 Analysis Results

Input file

Adhesive-Spring.inp : Using spring elemnt
Adhesive-I2D4-U.inp : Using interface element using uncoupled UGeneric material
Adhesive-I2D4-V.inp : Using interface element using uncoupled von Mises material
Adhesive-I2D4-C.inp : Using interface element using Cohesive material

E.8.3 Nonlinear Analysis of Composite Beam

As shown in the figure, the behavior of a simply supported composite beam was analyzed under two loading conditions. The upper and lower steel beams were modeled using beam elements, while the interface was modeled as BeamLink, RigidLink, or BeamJoint depending on the bond behavior. The following summarizes the analysis cases and notes for modeling:

Full Bond: Beam + BeamLink constraint
- The BeamLink constraint imposes the conditions
  \(\small v_{1} = v_{2}\), \(\small u_{1} = u_{2} + h\theta\), and \(\small \theta_{1} = \theta_{2}\).
No Bond: Beam + RigidLink constraint
- The RigidLink constraint imposes the condition
  \(\small v_{1} = v_{2}\).
- Note: For LC1 analysis, apply the same rotation to the upper and lower nodes.
Bond Behavior: Beam + RigidLink constraint + Spring
- The RigidLink constraint imposes the condition
  \(\small v_{1} = v_{2}\).
- A nonlinear shear spring in the form \(\small f = f(\delta)\) (where \(\small f\) is the spring force) is applied through a spring element with rigid arms, where the deformation is calculated as
  \(\small \delta = u_{1} - u_{2} + h_{1}\theta_{1} + h_{2}\theta_{2}\).
- Note: For LC1 analysis, apply the same rotation to the upper and lower nodes.

Figure E.8.7 Compsite beam

The element size was 50 mm, with 40 beam elements each for the upper and lower beams, and 41 spring elements. The BeamLink and RigidLink constraints also connected 41 pairs of nodes (see Figure E.8.7).

Figure E.8.8 Modeling

Figure E.8.9 Analysis Results of LC1

Figure E.8.10 Analysis Results of LC2

Input file

CB-F.inp : Full Bond
CB-N.inp : No Bond
CB-B.inp : Partial Bond (Bonded Case

E.8.4 Beam on Elastic foundation

An analysis was performed for a beam resting on an infinite elastic foundation subjected to a concentrated load and a distributed load (Figure E.8.11). For the case of a concentrated load, the results were compared with the analytical solution.

Figure E.8.11 Infinite elastic foundation

The numerical model was constructed with a sufficiently large length (600 m in this case) and modeled using beam elements and vertical springs (Figure E.15.12).

Figure E.8.12 Modeling

Figure E.8.13 compares the analytical and numerical solutions for the case of a concentrated load, showing good agreement. Figure E.8.14 presents the numerical solution for the case where a vertical distributed load is applied over a length of 200 m.

Figure E.8.13 Analysis Result of Case 1

Figure E.8.14 Analysis Reuslt of Case 2

□ Remark. Infinite beam on elastic foundation의 해석해

(1) Case 1 : Static concentric load

For an infinite beam on an elastic foundation subjected to a concentrated load, the governing equation is as follows.

\[ \small EIy^{''''} + ky = P\delta(x) \]

The solution satisfying the condition for the region \(\small x \geq 0\) is

\[ \small y(x) = \exp( - \lambda x)\left( C_{1}\cos{\lambda x} + C_{2}\sin{\lambda x} \right)\ ,\ where\ \ \ \lambda = \left( \frac{k}{4EI} \right)^{1/4} \]

Applying the symmetry condition and the equilibrium condition,

\[ \small \frac{dy}{dx}(0) = 0 \]

\[ \small 2\int_{0}^{\infty}{ky(x)dx} = P \]

yields the following solution:

\[ \small y(x) = \frac{P\lambda}{2k}\exp( - \lambda x)\left( \cos{\lambda x} + \sin{\lambda x} \right)\ ,\ where\ \ \ \lambda = \left( \frac{k}{4EI} \right)^{1/4},\ \ \ for\ x \geq 0 \]

(2) Case 2 : Harmonic load

When the beam is dynamically excited by a concentrated load, the equation of motion is given by

\[ \small EI\frac{\partial^{4}y}{\partial x^{4}} + m\frac{\partial^{2}y}{\partial t^{2}} + c\frac{\partial y}{\partial t} + ky = f(x,t),\quad y = y(x,t) \]

If harmonic excitation is applied at \(\small x = 0\),

\[ \small f(t) = F(t)\delta(x) = F_{0}\exp(i\omega t)\delta(x) \]

the dynamic response can be expressed as \(\small y(x,t) = \widehat{y}(x,\omega)\exp(i\omega t)\).
Therefore, the governing equation in the frequency domain becomes

\[ \small EI\frac{\partial^{4}\widehat{y}}{\partial x^{4}} + D(\omega)\widehat{y} = F_{0}\delta(x) \]

where

\[ \small D(\omega) = - m\omega^{2} + c\omega i + k = k\left[ 1 - \frac{\omega^{2}}{\omega_{n}^{2}} + 2\frac{\omega}{\omega_{n}}\xi i \right] \]

\[ \small \omega_{n} = \sqrt{\frac{k}{m}},\quad \xi = \frac{c}{c_{cr}},\quad c_{cr} = \sqrt{2mk} \]

Here, \(\small D(\omega)\) is the dynamic stiffness.
Following the same approach as for the static deflection, the frequency-domain solution \(\small \widehat{y}(x,\omega)\) is

\[ \small \widehat{y}(x,\omega) = \frac{P\lambda}{2k}\exp( - \lambda x)\left( \cos{\lambda x} + \sin{\lambda x} \right),\quad \lambda(\omega) = \left( \frac{D(\omega)}{4EI} \right)^{1/4},\quad x \geq 0 \]

Thus, the time-domain solution is

\[ \small y(x,t) = \widehat{y}(x,\omega)\exp(i\omega t) = \frac{P\lambda}{2k}\exp( - \lambda x)\left( \cos{\lambda x} + \sin{\lambda x} \right)\exp(i\omega t) \]

For arbitrary excitation, the solution can be obtained by applying the Fourier transform based on the above formulation.

Input file

elfound. inp

E.8.5 Simple Gap

The first example analyzes a structure with a spring and a gap, as shown in the figure, and compares the results with the analytical solution.

Figure E.8.15 Simple Gap Example

Figure E.8.16 Analysis Result at Node 2

Input file

gap1.inp

E.8.6 Beam on gapped spring (underconstruction)

Figure E.8.17 Analysis Model

E.8.7 Pounding During Earthquake (under construction)

Figure E.8.18 Analysis Model

References

Yuan, H., Wu, Z. S. and Yoshizawa, H. (2001) "Theoretical Solutions on Interfacial Stress Transfer of Externally Bonded Steel/Composite Laminates", Journal of Structural Mechanics and Earthquake Engineering, JSCE, No. 675/1-55, pp. 27-39
Zhishen Wu, Hong Yuan, Hedong Niu (2002), "Stress Transfer and Fracture Propagation in Different Kinds of Adhesive Joints", Journal of Engineering Mechanics, ASCE, Vol. 128, No. 5, May 2002., pp. 562-573
Salari, M.R., Spacone, E., Shing P.B., and Frangopol D.M. (1998). "Nonlinear Analysis of Composite Beams with Deformable Shear Connections." ASCE Journal of Structural Engineering, Vol. 124, No. 10, P. pp. 1148-1158.
M. Hetenyi, an arbor (1946) "Beams on elastic foundation", The university of michigan press.