Shakedown limit of elastic–plastic offshore structures under cyclic wave loading

ARTICLE IN PRESS Ocean Engineering 35 (2008) 1854–1861

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Shakedown limit of elastic–plastic offshore structures under cyclic wave loading M.J. Fadaee a,, H. Saffari a, R. Tabatabaei b a b

Civil Engineering Department, Shahid Bahonar University of Kerman, Iran Civil Engineering Department, Islamic Azad University, Kerman Branch, Iran

a r t i c l e in f o

a b s t r a c t

Article history: Received 12 December 2006 Accepted 11 September 2008 Available online 26 September 2008

This paper uses theorem of shakedown to assess the shakedown limit of elastic–plastic offshore structures. For this aim, an envelope of elastic response of the structure to cyclic loading cases is required. The shakedown limit is basically a valid collapse mechanism and can be quantified using yield line analysis. In this work, Melan theorem of shakedown (lower bound) is employed. Requiring simple elastic envelope and the domain defining yield lines only are the advantages of the Melan theorem. The shakedown analysis can be conducted by the finite element method (FEM), which is the main body of this paper. In order to evaluate the method of this paper, which is in fact combining the Melan theorem and the FEM, two steel offshore frames are analyzed using the proposed method and the results obtained are compared with the results of classical non-linear analysis method. & 2008 Elsevier Ltd. All rights reserved.

Keywords: Cyclic loading Elastic–plastic Shakedown multiplier Morison equation

1. Introduction The aim of the current research is to use the static theorem of shakedown to assess the shakedown limit of elastic–plastic offshore structures under cyclic wave loading. This method requires an elastic stress envelope and an admissible shakedown limit based on static theorem. The shakedown limit is basically a valid collapse mechanism and a method of quantifying such mechanism, using a yield line method, is described in the literature (Garcea et al., 2005). Determining the elastic stress envelope by the finite element method (FEM) analysis is the main body of this paper. Limited data are required from FEM analysis, namely, the normal moment along the yield lines to quantify the elastic stress envelope. The static theorem can then be used in conjunction with the elastic stress envelope to obtain the shakedown limit. Classical shakedown theories are based on the statical or kinematical theorems and currently represent one of the most important achievements of the theory of plasticity. Some aspects of the above-mentioned theories can be explained as follows. Shakedown quasi-static theory for elastic–plastic solids has been presented in the classical work of Koiter (Koiter, 1956). The theory is naturally extended to general cyclic loading processes (Corradi and Zavelani, 1974; Konig and Maier, 1981; Konig, 1987; Gro-Wedge, 1997). There are two mathematical approaches to attack a problem in elasticity as well as in plasticity. The first one is to solve a set of differential equations for the problem. The

 Corresponding author.

E-mail address: [email protected] (R. Tabatabaei). 0029-8018/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2008.09.003

alternative is to determine the extremum point of a certain functional. In the latter approach, we usually have two extremum theorems dual to each other: one is expressed in statical variables and the other in kinematical variables. In practice, with a finite element scheme in elasticity, the kinematical approach is usually preferred over the statical one because the admissible kinematical variables given in displacements are subjected to fewer constraints than the statical ones, which should satisfy a priori the equilibrium equations. In addition, with dynamic processes, one cannot avoid the kinematical variables. In plastic limit analysis, the kinematics’ method of set up of admissible collapse mechanisms involving yield lines and curves is successful in solving many practical problems (Capurso, 1974; Polizzotto et al., 1993). However, in shakedown analysis, Koiter’s original kinematic theorem is difficult to use because it is complex and involves time integrals over loading processes. Recently, reduced forms of the kinematic theorem, which is simpler than the original one, have been constructed to solve various practical problems (Janas et al., 1995; Yaung-Gao and Zhang, 1995; Pham, 2001). Later, there was indicated in the last paper, the results can be a restricted class of problems. Their study addresses the more general class of problems, a deeper insight into the extensions of the classical theory to material with hardening effects, finite displacements, thermal and dynamics effects. (Stein et al., 1987; Ponter and Karter, 1997; Ponter and Engelhradt, 2000; Yan and Nguyen, 2001; Borino and Polizzotto, 1995). In this paper, the elastic stress envelope obtained by the finite element method is used to evaluate the shakedown safety factor for elastic–plastic offshore structures. The Morison equation is adopted for converting the velocity and acceleration of fluid

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particles into resultant forces applied on the structure members oriented arbitrarily. Then, the forces generated by a large number of periodic wave loading with different wave heights, periods and directions of propagation which all occur at the same time are estimated using regular wave theories. The two simple cases of offshore structures are used here for indicating the details of implementation and the capability of the method. Comparing the results with classical non-linear analysis output validates the results obtained.

2. Shakedown theorem and incremental collapse The phenomena of shakedown and incremental collapse were first proven theoretically by Melan (1938). Melan formulated the lower bound or the static theorem of shakedown. The static theorem of shakedown is such that shakedown will occur provided that at all times the stresses obtained from an elastic analysis satisfy the following: Shakedown theorem are stated based upon several assumptions concerning loadings intervals and corresponding stresses. According to the common structural codes, it is assumed that the external time-dependent load P(t) is a combination of the reference loads, Pi with a specific variation range defined by the following relationship PðtÞ ¼

p X

ai ðtÞPi amin pai ðtÞpamax ; 8t i i

(1)

1855

plastic strains increase after the first cycle of a specific load combination due to the waves such that in any cycle loading, P(t), the behavior of the structure is still elastic. This can be stated by the following relationship  Z 1 Z sðtÞT _ P ðtÞ dv dto0 (5) t¼0

B

in which s(t) is the stress component, _ P ðtÞ is the kinematics plastic strain component produced due to loading process, P(t), in any cycle. The time t ¼ 0 indicates the non-deformed initial state eP(0) ¼ 0. The numbers of the plastic hinges become enough for the structure shakedown in an incremental manner. It is obvious from Eq. (5) and from the above-mentioned definition that the shakedown happens when at least in one cycle of the loading combination and at a specific time, the domain of the residual stresses in the structure can be stated as follows, f ðlse ðtÞ þ sres Þp0

8se ðtÞ 2 stress domain

(6)

e

in which s(t) and s (t) are the domain of the total stress and the domain of the elastic stress due to the loading, P(t), respectively, which are in equilibrium condition. Therefore, their difference is equal to the residual stress, sres. Now, the shakedown analysis of the offshore structure under specific combination of nodal loading equivalent to the wave loads is conducted and the shakedown load multiplier is calculated. The multiplier more than unity means the corresponding wave causes the shakedown of the structure (Fig. 1).

i¼1

in which ai(t) are the multipliers that are defined in a polygon region by (Polizzotto, 1994). In this region, the load multipliers have been defined for elastic state, collapse mechanism and plastic state. In fact, based upon Eq. (1) it can be said that the real loads applied on the structure have been resulted from some unknown factors and they vary between a minimum and maximum value during the structure life. Therefore, in the shakedown analysis, the domain of the actual loads is obtained by multiplying the nodal reference loads Pi by the multipliers ai(t) and has specific minimum and maximum limits. The stress domain s(t) can be defined as a combination of elastic stresses, se and residual stresses, sres in the following form

sðtÞ ¼ se ðtÞ þ sres

(2)

The structure shakedown happens at a load multiplier l41 concerning the loading domain aiminpai(t)paimax such that the following relationship is satisfied f ðlse ðtÞ þ sres Þp0

(3)

where f is a yield function that can be simply defined by the yield theories. The domain of the residual stresses sres is time independent and the structure is still in equilibrium condition. The maximum value of the multiplier l that satisfies Eq. (3) is called the ‘‘shakedown safety factor’’ or, in summary, ‘‘shakedown multiplier’’ and is indicated by la symbol. From the abovementioned relationships, it is clear that the static theorem formulation is based upon the stresses domain which defines a lower bound for the load multiplier l. Usually, in practical problems, the static’s theorem is stated in an optimization problem form as follows, e

res

la ¼ max l : f ðls ðtÞ þ s Þp0

4. Shakedown loading domain based upon airy linear wave theory Consider a pipe element with arbitrary direction in xyz system as shown in Fig. 2. Now, if the movement of the fluid consisting of horizontal component of velocity, v, vertical component of velocity, u, horizontal component of acceleration, ax, and vertical component of acceleration, ay, is known, the force resulted from the waves applying on the pipe element can be calculated. In cylindrical system, f and c indicate the direction of the cylinder. The component of the velocity of the water perpendicular to the cylinder axis, V, can be found using the following relationship (Noorzaei et al., 2005) V ¼ ½u2 þ v2  ðcx u þ cy vÞ2 1=2

(7)

The components of the velocity in x, y and z directions are computed from the following relationships un ¼ u  cx ðcx u þ cy vÞ vn ¼ v  cy ðcx u þ cy vÞ wn ¼ cz ðcx u þ cy vÞ

(8)

c

L

H

SWL

(4)

h 3. Occurring shakedown in an offshore structure y=0

Based upon the shakedown theorem explained in Section 2 of this paper, the offshore structure shakedown will happen if the

Fig. 1. Definition sketch for progressive waves.

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QMj QVj

Unx Unz

Wn b

Uny φ

Q2

a

l

l

ψ Q1

a

QVi Fig. 2. A typical tubular element in cylindrical coordinate.

QMi Fig. 3. Fixed end forces of a member under wave loading.

where cx ¼ sin f cos y; cy ¼ cos f; cz ¼ sin f sin y

(9)

Also, the components of the wave acceleration in x, y and z directions can be determined as (Sarpkaya and Isaacson, 1981), anx ¼ ax  cx ðcx ax þ cy ay Þ any ¼ ay  cy ðcx ax þ cy ay Þ anz ¼ cz ðcx ax þ cy ay Þ.

(10)

The components of the force applying per unit length of the cylinder in x, y and z directions are calculated as follows, 1 pD2 anx rC D Dun þ rC I 2 4 2 1 pD any f y ¼ rC D Dvn þ rC I 2 4 2 1 pD anz f z ¼ rC D Dwn þ rC I 2 4 fx ¼

(11)

in which r is the density of the water, D is the diameter of the pipe element of the structure and CD and CI are the drag coefficient and the inertia coefficient of the water particles, respectively. The force perpendicular to the unit length of the pipe element is 2

2

2

f ¼ ðf x þ f y þ f z Þ1=2

(12)

The sign of the force is chosen concerning the signs of fx, fy and fz components. So, typically an offshore structure element is under a non-uniform distributed loading along the length of the member based upon Eq. (12). Moreover, the equivalent nodal forces at the two ends of the member can be calculated using the equilibrium equations. The resultants of the forces can be determined by integration as Z Z Z Fx ¼ f x ds; F y ¼ f y ds; F z ¼ f z ds (13) s

s

s

where s is the loading length along the element axis. Since the load due the fluid varies nonlinearly along the element, integrations Eq. (13) are very complex and time consuming, so for calculating the forces, a simple model for converting the distributed loads resulted from the sea waves into the equivalent nodal loads has been presented (see Fig. 3). In the proposed model, it is assumed that the load varies linearly in several parts along the element. First, the velocity and the acceleration at the end of each element have been determined and their variation along the element has been assumed linear. Then, the values of the velocity and acceleration at the middle point of the element have been obtained using linear diagram of the load variation

assumed and compared with the values at the two ends. If the difference between the value of the middle point and the values at the ends is more than 5%, then the element will be divided into two parts such that the loading on each part varies linearly. This process continues until the difference becomes less than 5%. So, the intensity of the loading Q1 and Q2 on each part is equal to the maximum value of two ends and is considered uniformly distributed. Q1 and Q2 are the intensities of the loading resulted from cyclic sea waves in terms of variable parameter ot. They must be converted into equivalent nodal end loads in local coordinate system first, and then converted into global coordinate system using a transform matrix. The vector of the equivalent nodal loads for an element in local coordinate system is as f ðtÞ ¼

n

QM i

QV i

QMj

QV j

oT

.

(14)

Also, the equivalent nodal loads in global coordinate system can be determined using the coordinate transformation matrix, Rb, as follows, PðtÞ ¼ Rb f ðtÞ

(15)

The coordinate transformation matrix is discussed in Dawson (1983). The relationships in local coordinate system are determined as detailed in Appendix 1 for calculating the nodal end loads.

5. Elastic response of the structure to cyclic wave loading Consider a prismatic pipe element to be used for analyzing the offshore structure as shown in Fig. 4. In the modeling, the near and far ends of the element are indicated by i and j, respectively. In the member coordinate system, {x,y}, the x-axis is the axis of the member, and the components of the displacement is stated by the movement perpendicular to the cross section of the member in xy plane, such that u(x) is the axial displacement, w(x) is the transverse displacement and y(x) is the rotation component. Also, the stresses are corresponding to the axial force, N(x), shear force, V(x) and bending moment, M(x), applying at the cross sections of the member. The axial and shear components are constant, but the moment varies linearly (Casciaro and Garcea, 2002). Therefore, details of the FEM formulation is shown in Appendix 2.

ARTICLE IN PRESS M.J. Fadaee et al. / Ocean Engineering 35 (2008) 1854–1861

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φe uj

φs

ui

Wj

θj

φe

φa

φs

θi

Fig. 4. Kinematics parameters and natural modes for element of structure.

6. Admissible shakedown domain Defining an admissible domain for yield condition and for forming the plastic hinge in the member is one of the main points in shakedown analysis. For simplicity of the analysis, the bending moment, M, is considered for plastic state only. So, the problem is defined as follows, þ M¼ y ðxÞpMpM y ðxÞ

(16)

in which xA[0,y,l] and My+(x) and My(x) are the positive and negative yield moments, respectively. The response of the basic external elastic moments is as follows, M E ðxÞ ¼

P X

ai MEi ðxÞ;

amin pai pamax i i

(17)

i¼1

Therefore, the maximum and minimum of the responses of the elastic moments are as follows, M E ðxÞ

E

Mþ E ðxÞ

¼ minðM ðxÞÞ;

E

¼ maxðM ðxÞÞ

(18)

Now, using the values as determined in the above, the admissible shakedown analysis domain for a member of an offshore structure with i and j ends can be easily defined by the following relationships  þ þ M yi  lM Ei pM i pM yi  lM Ei

(19)

 þ þ M yj  lM Ej pM j pM yj  lM Ej

(20)

The following relationship can be found as, M i ¼ ðms þ me Þ=2;

M j ¼ ðms  me Þ=2

(21)

Finally, the largest admissible load multiplier is as follows, ( þ ) Mym  M  ym la ¼ min (22)  Mþ Em  M Em where m relates to the control at the end section of all elements of the offshore structure. A collapse mechanism for the plastic rotation, y_ m , at the section m is defined first. The plastic hinge will be formed under one of the following conditions. 8 max þ < m ¯ m þ Mm ¼ M pm if y_ m 40 (23) min :m ¼ M  if y_ m o0 ¯m M m

M+pm

pm

M pm

where and are the maximum and minimum values of plastic moment, respectively, and parameter m ¯ m is the residual

moment at the sections {m ¼ 1, 2, 3, y} and are under equilibrium condition. Applying the material of Section 2 of this paper and using the virtual work theorem, the shakedown analysis is done through the following relationship. 9 8 P þ _þ  _ = < m¼1 ðM pm ym þ M pm ym Þ 8m (24) la ¼ min P þ  þ  : ðM ðxÞy_ þ M ðxÞy_ Þ; m¼1

E

m

E

m

Now, considering the equivalent nodal loads due to the waves obtained in Section 4 of this paper and applying quadratic programming optimization algorithm the relationship Eq. (24) is controlled for all the members of the structure and finally the shakedown multiplier, la, is computed. It must be noted if the multiplier of the shakedown load is more than unity, the multiplier obtained will be the same as a collapse load multiplier.

7. Domain of the shakedown unsafe waves For any offshore structure, it is necessary to find a repeating wave loads domain under which shakedown of the structure certainly happens. Here, after shakedown analysis of the structure, if the multiplier of the shakedown load is more than unity, the corresponding load combination will be considered as critical shakedown load. In this study, the domain of the waves which cause shakedown is called ‘‘unsafe waves domain’’ for offshore structure shakedown. Now, having different water depth to wave length ratios, h/L, and different wave height to wave length ratios, H/L, for the critical shakedown load waves, the graph of the domain of the shakedown unsafe waves can be drawn.

8. Numerical studies 8.1. Elastic–plastic modeling of the elements In each of the following example, the beams and columns are modeled by standard frame elements. Elements are assumed to behave linearly except at pre-defined plastic hinge locations, which are located at 5% of the length from each end of each member. Plastic hinges are assigned a 5% strain hardening ratio and are assumed to fail completely at a rotational ductility. Floors are assumed to act as rigid diaphragms distributing their mass uniformly on the supporting beams. The hysteretic properties of K elements are based upon a trilinear idealized hysteresis curve as shown in Fig. 5a. In order to

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Fig. 5. Modelling of a K element: (a) idealization of hysteresis data and (b) typical skeleton curves.

Fig. 6. Modelling of a X bracing element: (a) idealization of hysteresis data and (b) typical skeleton curves.

come up with a series of simple, idealized K elements properties, it is assumed that both strength and stiffness could be scaled in the same way for K element size, so that only the yield force vary between section, not the yield deformation. This is not strictly correct, but the error, thus, introduced is likely to be small. Fig. 5b shows the skeleton curves for a typical K element. A similar approach is adopted for modelling the X bracing elements, except that the initial idealization is simply to elastic–perfectly plastic. Typical curves are shown in Fig. 6.

1

2

3

l

4

5

l=6943 Cm D=120 Cm Column Diameter D1=60 Cm Beam & Bracing Diameter t=2.5 Cm thickness of D t1=1.25 Cm thickness of D1

F y = 2400

Kg Cm

E = 2.03 x 10

6

Kg Cm

l

7

8.2. Example 1: an offshore platform frame with X bracings

x y 6

8 l

Fig. 7. Offshore platform frame considered in numerical example 1.

0.3 0.25 0.2

H/L

The material properties and dimensions of the X bracing of offshore platform structure are shown in Fig. 7. All members are considered as elastic-hardening materials in the analysis. The forces caused by a sine wave with specific characteristics are determined using the Airy wave theory and converted into equivalent nodal loads. Afterwards, by the shakedown analysis using the static Melan theorem and an optimization algorithm, the shakedown multiplier has been found for the specific wave. If the shakedown multiplier is more than or equal to unity, the wave is critical and its characteristics will be recorded. This process is repeated for all the waves that possibly encountered the structure. At the end, the critical region in which the shakedown happens is assigned on a graph. The details of the information obtained for this example is shown in Fig. 8. As it can be seen, the graph has been drawn for all the values of ratios h/L and H/L obtained from the Airy theory. The non-hashed region shows the waves that if are repeatedly applied on the structure, the collapse mechanism forms and the structure shakedown happens. So, a part of the region is critical. For other values, the region is called ‘‘structural safety domain’’. The dashed member in Fig. 12 indicates forming the first member collapse mechanism in shakedown analysis.

0.15

Structural safety domain

0.1 0.05 0 0.00

0.04

0.05

0.09

0.16

0.1h/L Fig. 8. Safe region for shakedown in example 1.

0.23

0.30

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Base shear in X-direction (kN)

Validations of presented methodology are verified using classical non-linear analysis. The sine wave having H/L ¼ 0.3 and peak acceleration equal to g is applied along the X-direction. The nonlinear response history analysis (RHA) is carried out using the DRAIN-2DX computer program (Powell et al., 1993). The base shear in X-direction versus time of the structure subjected to the sine wave obtained by RHA is shown in Fig. 9. Further, Fig. 10 displays the capacity curve of the structure estimated by nonlinear static analysis (NSA). The comparison of the base shear ratio as calculated by the present method with those determined by RHA and NSA under the ultimate condition is shown in Fig. 11 which illustrates that the error will be increased as H/L ratio of the input sine wave load is increased. The most notable reason is that in the shakedown approach, the structures are responding nearly

500 400 300 200 100 0 -100 0 -200

5

10

15

20

-300 -400 -500 Time (sec)

Fig. 9. Non-linear response time histories of example 1 under sine wave (H/L ¼ 0.3).

400

1859

elastically. On the other hand, in the present method, the energy dissipation is not considered. In this example, the shakedown analysis algorithm takes 20% central processing unit (CPU) time less than the RHA method processes. 8.3. Example 2: an offshore platform frame with K element The second example studied in this paper is a usual and practical offshore structure with K element under sea waves loading as shown in Fig. 13. The geometric and the material characteristics of the structure members are also indicated in the figure. The materials of the structure members are of the elastic–plastic type having a hardening part in the moment– rotation curve. The finite element model of the structure including members and nodes has been prepared first, and then the wave loading defined by the Airy theory has been applied on the structure. In the next step, the shakedown multiplier for the equivalent nodal load due to the maximum wave amplitude has been calculated using the shakedown analysis. Then, the unsafe region for critical shakedown load multipliers has been shown in Fig. 14. As mentioned before, the non-hashed region indicates the waves’ characteristics that cause the structure shakedown. The response time history of the original structure is obtained using the arithmetic summation of the modal response time histories and the capacity curve subjected to sine wave (with H/L ¼ 0.3 and peak acceleration equal to 0.1g) is estimated by NSA (see Figs. 15 and 16). The comparison of the base shear ratio calculated by the present method with the results determined by NSA under ultimate condition is shown in Fig. 17. The CPU time in the present method has been 30% less than RHA processes.

9. Conclusion

Base shear (kN)

350 300 250 200 150 100 50 0 0

0.005

0.01

0.015

0.02

Displacement at joint 2 (m) Fig. 10. Capasity curve of example 1 obtained by NSA.

There is a strong trend towards the use of FEM in shakedown approach for offshore structures. The availability and powerfulness of the innovative FEM formulation is a convenience for shakedown theorem. It is important to know that a shakedown approach will never be able to completely replace a non-linear dynamic analysis; nevertheless, a methodology is searched to obtain response information reasonably close to that predicted by the nonlinear time history analyses. The innovative shakedown approach is, therefore, shown to constitute an extremely appealing collapse mechanism for offshore structural assessment, fully in line with the recently introduced collapse mechanism trends in the field of ocean engineering. Further, the equivalent nodal loads 1

2

0.3

l

4 0.25 0.2

5

0.15

7

0.1

l

H/L

3

VS/VN

0.05

x

VS/VR

y 6

0 0.9

1

1.1

Fig. 11. Comparision of base shear ratio for example 1.

1.2

8 l

Fig. 12. Collapse mode in example 1.

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4 1

2

l

wave load

l=7000 Cm D=30 Cm Column Diameter D1=20 Cm Beam & Bracing Diameter t=1.5 Cm thickness of D t1=1.0 Cm thickness of D1 Kg

7 5

Cm 2 Kg 2 Cm

l

3

6

8 l

Fig. 13. Offshore structure considered in numerical example 2.

0.30

0.3

0.25

0.25 0.2

0.15

H/L

H/L

0.20

0.15

0.10

Structural safety domain

0.1

0.05 0.00 0.01

0.05

0.08

0.14

0.21

VS/VR

0.30

0

0.1h/L

0.9

Fig. 14. Safe region for shakedown in example 2.

Base shear in X-direction (kN)

VS/VN

0.05 0.03

6.0 4.0 2.0 5

10

15

20

-4.0 -6.0 -8.0 Time (sec)

Fig. 15. Non-linear response time histories of example 2 under sine wave (H/L ¼ 0.3).

due to the maximum of the sea waves have been used for shakedown analysis of the structure in order to find the shakedown load multiplier. It has been shown by the numerical examples that the simple proposed method in this paper to estimate the wave loading on members which uses shakedown approach algorithm decreases the time and the cost for non-linear analysis of huge offshore structures in addition to having the ability of predicting the collapse mechanism. It is recommended that the offshore structures designed, manufactured and erected previously must be evaluated for shakedown and if the unsafe waves concerning shakedown may happen at the site that those structures are located, they must be strengthened in this regard. The method developed in this paper is suitable for large offshore structures because of its simplicity. Also, the proposed method can help the designers for designing the geometry of the structure and choosing the materials characteristics.

8

Base shear (kN)

7

Appendix 1

6 5 4

QM i ¼ M 1 þ M 2

3

QM j ¼ M 3 þ M 4

2

The horizontal shear forces are as

1 0 0

0.002

0.004

0.006

0.008

0.01

QV i ¼

2alQ 1  2M 3 þ 2M 1  Q 1 a2 2M 2  2M4 þ Q 2 aða þ 2bÞ þ 2l 2l

QV j ¼

2M 3  2M1 þ Q 1 a2 2M 4  2M 2 þ 3a2 Q 2 þ 2l 2l

Displacement at joint 2 (m) Fig. 16. Capasity curve of example 2 obtained by NSA.

1.1

Fig. 17. Comparision of base shear ratio for example 2.

8.0

0.0 -2.0 0

1

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in which l is the length of the member. Moreover, the following values are used for computing the fixed end moments at the two ends. Q al M 1 ¼ 1 mð3m2  8m þ 6Þ 12 Q2

M3 ¼

Q 1 al 2 m ð4  3mÞ 12 Q2 2

a ; l

½8a3 ðl þ 3bÞ  a3 ð4l  3aÞ

e¼la

Appendix 2 ma me 1 ; V½x ¼  ; M½x ¼ ðms þ me ð1  2xÞÞ 2 l l ma ¼ Nl; ms ¼ M i þ M j ; me ¼ M i  M j N½x ¼

where l ¼ xjxi is the member length and x ¼ (xxi)/l is a dimensionless coordinate which varies in the interval [0,1]. The natural strains can be defined as

fa ¼ ðuj  ui Þ=l;

fs ¼ ðyj  yi Þ=l;

fe ¼ ðyj  yi Þ=2  ðwj  wi Þ=l

The strain energy can be found as 8 9T 2 9 38 f = > fa > = ka 0 0 > < a> 1< 60 k fs 07 Pb ¼ 4 5 fs s > > 2> :f > ; 0 0 ke : fe ; e where ka ¼ EAl;

ks ¼ 4EI ; l

12EI ke ¼ lð1þ bÞ ;

12EI b ¼ GkAl 2

The components of the displacement field for the two ends of the member are ub ¼ f ui

wi

yi

uj

wj

The natural strains and field can be indicated as 2 C 9 8 6l > 6 = < fa > fs ¼ T b ub ; T b ¼ 6 6 0 6 > ; :f > 4 S e  l

yj gT the components of the displacement S  l 0 C l

0 1=2 1=2

C l 0 S l

S l 0 C  l

3 0

7 7 1=2 7 7 7 5 1=2

S ¼ Sin a C ¼ Cos a

in which a is the angle between the global coordinate system and the local coordinate system. Finally, the strain energy can be computed by 1 2

ka

0 ks 0

0

3

T 07 5T b ;

ke

9 8 > = < ma > F b ¼ T Tb ms > ; :m > e

References

The parameters, m and e are defined as m¼

2

3

½e3 ð4l  3eÞ  b ð4l  3bÞ 2 12l

12l

The elastic stiffness matrix and the vector of the internal forces are determined as

6 K b ¼ T Tb 4 0 0

M2 ¼

M4 ¼

1861

Pb ¼ uTb K b ub  uTb F b .

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