Nonlinear Analysis with Tri-linear M-Y Curve

In this section the formulation of nonlinear analysis for tri-linear type of connection is presented. Fig.

(1) shows such a typical relation. A tri-linear relation usually fits better on an M-Y curve. It can also model strain-hardening of a typical connection. This type of relation may be applicable for both negative and positive moment directions

As shown in Fig. (4) the tri-linear relation may be characterized by four parameters: M1, Mlim, R0 and R1. In which Mlim is the limit of bending moment capacity, R0 and R1 are initial and secondary flexural stiffness of the connection and Mi is the first yield in the M-Y curve. Similar to the previous procedure for formulation of the problem, first the case of negative moment is considered.

Depending on the value of moment or rotation in a connection, the M-Y relation may coincide one of the three linear relations shown in Fig. (4).

If в < вл, then the connection behaves linearly. In this case: Mr= Mi and there will be no need for external virtual moment.

If вл < в < ві2 , the actual value of Mi for a given value of Y can be obtained from Eq. (12).

M-= Mh. + (в, -ви)*Яи (12)

From linear analysis we have: Mi = R0iet and M1 = ROi0l, therefore it looks possible to substitute for values of Yi and Y1i to find:

R

m г )* ^ Rn

Eq. (13) stipulates that in the connection with reduced stiffness, only partial part of the moment in excess of M1 is stored and some part of the excess moment should be distributed in the structure.

Л

From the above explanation it is understood that the (Mi-M-)*( 1 – R1i /R0)part of the excess moment should be distributed in the structure. However any moment redistribution from connection into the structure not only affects the value of rotations of connection and corresponding moments in connected members but also affects the moment distribution in the structure which in turn affects again the value of excess moment in the connection. The study on this mutual moment redistribution between softened connection and the structure shows that there is a reciprocal relation as in Eq. (j3) between remained moment in the connection and other parameters such as the moment distribution factor mii, the ratio of stiffness of member to the connection (EI/L)/R0 and the rate of stiffness softening R0/Rj.

remained Moment in connection i = yi XMoment in excess of M1

1 ! ,R-0 ,w4EI/L.

— =1 + mu( – -1)( )

Y R1 R0

in which mu is the measured moment at connection i due to application of a unit moment at i. The statement in Eq. (14) can be mathematically written as the following equation for the softened range of behaviour of the connection.

M – = Mi + £+ X+ -(Xj + X2j)) >M – +AMi * Yi

j=1

j=1,j *i

The value of AM in the above equation includes all transferred moments from all virtually loaded connections onto the connection in question i. e.

Substituting for AM from Eq. (16) into Eq. (15) and rearranging for unknowns gives the yield condition for the case of stiffness softening in the connection as follows:

muX 1+ + (1 – Yt) £ mj(X + + X + – (X – + X -j )) > (M – – Mi )*( 1 ~Yt)

j=1,j*i

Similarly when 6t > 6t2 we should add the external virtual moment Xi2 to the connection to reduce the moment to Mlim. In this case the following relation can be written between resultant moment in joint i and limit moment:

M – = Mi + £mj(X+ + X+. -(X1- + X-j)) >M-m (18)

j=1

It should be noted that, although Xi1 andXi2 as shown in Fig.(4), are independent virtual moments, they can not exist simultaneously. Since it is not known a priori that how much is the, both of

Eqs.(17&18) should be considered in the solution of the problem. In general the QP problem for calculation of unknown X’s become as follows:

Subject to : muX + + (1 – Y ) J my(X + + X + – (X – + X – )) > (M – – Mi )*(1 – Y )

j=1,j*i

£ my((X +j – X – ) + (X 2j – X 2j )) > M-m – Mi ;i = 1,…,n

j=1

Similar QP problem can be established for reverse direction i. e. the case of a tri-linear connection in positive moment direction and unlimited linear relation in negative moment direction. If R0 , R1 , M1 and Mlim properties in positive direction are assumed to be the same as their counterpart in negative direction, the following QP problem can be obtained. However one may use different properties for two different directions of moments.

Subject to : muX 1 + (1 – yt) £ m y(X + + X 2+ – (X ~ + X – j )) > (M t – M + ) * (1 – yt)

j=1,j*t

+X t2 [M+mj – Mt + £ m, j((X+j – X-j) + (X+y – X-j))] + (X+ + X+2) X (Xt1 + X-)

j=1

Subject to: m aX+ + (1 – Y ) £ my (X + + X 2+. – (X-j + X y )) > (M-j – Mt )*(1 – Y )

j=1,j*t

UX1- + (1 – Y ) £m j (X+j + X2+j – (X-j + X2-j )) > (Mt – M+ )*(1 – y )

j=1,j*t

It is to be noted that in general a connection is separate than a member and its role should be modeled as a torsional spring with rigidity of Ro, Ri and 0 at the three ranges of rotations. In this example the connections in 2 and 3 had to be modeled as in Figure 7-a. This required special analysis software. Instead the model in figure 7-b which can be analyzed by most of commercial softwares was used. In this model torsional rigidity of the members 2-6 and 3-7 are equal to the flexural stiffness of connection. Note that in nodes 1 and 4 there is no connection and the capacity of the structure at these nodes is determines by capacity of the members.

To solve the problem we need to perform a linear elastic analysis for the applied loads and do some analysis for unit moments. The number of virtual unit moments depends on the number of potentially nonlinear points in the structure. Here we assume that all nodes will have nonlinear behavior. However the behavior of joints 1 and 4 will be assumed to be the same as elastic-plastic behavior of members. The moment at midspan does not seem to exceed its proportionate limit. As a result unit moment has not been applied there. Therefore there will be four virtual loadings. The results of aforementioned analyses have been reported in Table 3.

Table 3: Results of Analysis of Frame of Example 2 for Various Types of Loadings

Moment

Full

External

Load

Unit M at 1 in

Unit M at 2 in

Unit M at 3 in

Unit M at 4 in

Position

member 1-2

member 2-3

member 2-3

member 3-4

Joint No. 1

-92.815

1.0-0.8503

0.12693

-0.10811

0.32734

Connection 2

7.726

0.5476-0.5

1.0-0.85089

0.4961-0.5

-0.00362

Midspan

28.256

0.00353

0.07260

0.07694

-0.03709

Connection 3

-11.213

-0.04054

0.4961-.5

1.0-0.84222

0.46202-0.5

Joint No. 4

27.206

+0.16367

-0.04828

-0.05064

1-0.31091

According to the results in Table 3, a QP sub-problem, can be established as follows. In the following QP problem a combination of Eq.(11) and Eq.(21) is used. This is because the joints 1 and 4 have bilinear behavior and obey the rules of Eq.(11) and joints 2 and 3 have tri-linear behavior and obey Eq.(21). In addition since in this simple problem we know the direction of applied virtual moments, it is evident that X – = X +1 = X +2 = X — = X -2 = X + = 0. Therefore only those virtual moments that potentially are nonzero have been participated in the formulation. Because of this decision, only the complementary condition on X± XX±2 = 0. has been considered.

Minimize {X+ [(Mі – Mliml) + m11X+ – m12(X12 + X22 ) + m13 (X+ + X +3) – m14X4 ) +

л +

X12[(1 — Y2 )(M 1,2 — M 2 ) + m22X12 — (1 — Y)(m21X1 — m23 (X31 + X32 ) + m24X4 )] +

л+

X22 [(Mlim,2 — M 2 ) — [m21X1 — m22X22 + m23 (X31 + X32 ) — m24X4 )] + (X 12 *X22 ) +

X13 [(1 — Ї3 )(M 3 — M13 ) + m33X13 + (1 — Ї3 )(m31X1 — m32 (X21 + X22 ) + m34X4 )] +

X23 [(M 3 — Mlim,3 ) + m31X1 — m32 (X21 + X22 ) + m33X23 — m34X4 )] + (X 13 * X23 ) +

л+

X4 [(Mlim,4 — M 4 ) — m41X1 + m42 (X21 + X22 ) — m43 (X31 + X32 ) + m44X4 ]}

Subject to : m11 X + — m12 (X-2 + X-2 ) + m13 (X^ + X +3) — m14X – ) > MUm1 — M1

л+

m22 X12 – (1 – Y2 )(m21X1 + m23 (X 13 + X23 ) – m24X4 )) > (1 – Y2 )(M 2 – M12 )

л+

– m21X1 + m22X22 – m23 (X13 + X23 ) + m24X4 >M 2 – Mlim,2

m33X13 + (1 – Y3 )(m31X1 – m32 (X 21 + X 22 ) – m34X 4 > (1 – Y3 )(M13 – M 3 )

m31X1 – m32 (X21 + X22 ) + m33X23 – m34X4 > Mlim,3 – M 3

л+

– m41X1 + m42 (X21 + X22 ) – m43 (X31 + X32 ) + m44X4 >M 4 – Mlim,4

X -2 * X -2 = 0 X +3 * X 2+3 = 0

In this sub-problem m;j values are given in Table 3. Other parameters are as follows:

M+m,1 = – MНЩ1 = M+m4 =-M~m4 = 50 KN. m M +2 = – M u = M +3 = – M u = 28 KN. m

ML,2 = – Mjim,2 = ML,3 = – MUm,3 = 42 KN. m Y2 = 0.788776 and y3 = 0.7593655

Solution of the problem yields the following results:

X+= 574.1396 Xi2l = 54.6801 X+1 = 10.6389 X-= 107.0615

According to these results evaluation of internal forces in members is quite simple. Table 4 shows the calculation of final nonlinear response for moments.

It can be seen from the Table 4 that the position of vertical point load (Midspan) is still in its proportionate range and therefore, as it was foreseen, there is no need to apply external virtual moments. On the other hand except Node No. 1 it did not seem that other nodes can reach to nonlinear stage. But results show that Node No. 4 has reached to its plastic limit; and the moments at connections No. 2 & 3 are a little beyond the first yield of the connection. It is obvious that if the load is increased, the moments at connections 1 and 4 will remain constant and the moments of Midspan, Node 2 and Node 3 will increase.

Table 4: Calculation of nonlinear response of Example 2, Proposed method.

Effect of Xj at joints and connections

Values of Virtual Moments

Total Moment

Moment Position

X1

X 2-

X 3+

X 4

M + X + + X 2­+ X 3+!+ X –

574.1396

54.6801

10.6389

107.0615

Node No. 1

85.9506

-6.9404

-1.1502

-35.0454

-50.0000

Connection No. 2

27.3277

-8.1531

-0.0415

3.8752

30.7340

Midspan

2.0254

-3.9699

0.8186

3.9705

31.1010

Connection No. 3

-23.2768

0.2134

1.6787

4.0658

-28.5319

Node No. 4

93.9691

2.6389

-0.5387

-73.7753

50.0000

Conclusion

In this paper, a new, non-iterative method of nonlinear analysis was proposed. In the proposed method to make internal moments follow their nonlinear moment-rotation curves, some virtual moments (that are primarily unknown) were imposed onto the structure at semirigid connections. To find the unknown virtual moments, a quadratic programming problem was formulated and solved. In the proposed method, the exact nonlinear response of structure including internal forces and moments of members are calculated by employing the values of virtual moments and using the superposition principle. Compared to the classical methods of nonlinear analysis, the following preferences can be mentioned for the proposed method.

• In the classical methods, the nonlinear analysis is done through iterative procedures which consist of modification of the stiffness matrix of structure and/or load vector, but in the proposed method the nonlinear analysis is performed in one step without change in the initial model and stiffness matrix of the structure or its load vector.

• the final result of iterative methods are somewhat dependant on the start point and the convergence criteria while in the proposed method it is obtained by solution of a Quadratic programming method and therefore not only it does not require any initial point, but also it does not require convergence criterion and mathematically saying it gives exact results.

• To increase the accuracy of the results in the classical methods it is necessary to decrease the incremental loading and tighten the convergence criterion. This increases the number of required analysis. However in the proposed method the results are exact and only preventing round off error in calculations increases the accuracy.

The method is capable to model semirigid connections with multilinear moment-curvature relations. The formulation of the problem for bilinear and trilinear moment-curvature relations was presented. Two examples were solved to demonstrate the robustness, capability and validity of the method. It was shown that the method not only can be used for nonlinear connections but also provided that the critical sections are pre-specified, it can do the plastic analysis in a flexural frame as well.

References.

Abu Kassim, A. M. and Topping, B. H.V. (1985). "The theorems of structural variation for linear and nonlinear finite element analysis", Proceeding of the second international conference on Civil and Struc. Eng. Comp. CIVIL-COMP 85, Vol. 2, pp. 159-168

Crisfield M. A., (1991). "Nonlinear finite element analysis of solids and structures", Volume 1- Essentials. John Wiley & Sons, Chichester, U. K.

De Donato, O. (1977) "Fundamentals of elastic-plastic analysis", Engineering plasticity by mathematical programming, M. Z. Cohn and G. Maier, editors. Pergamon Press, New York, N. Y. pp. 325-349

Katsuki, S., Frangopol, D. M. and Ishikawa, N. (1993). "Holonomic elasto-plastic reliability analysis of truss systems. II: Applications." Journal of Structural Engineering, ASCE, Vol. 119, No. 6 pp. 1792­1806

Majid, K. I. and Celik, T., (1985). "The elastic-plastic analysis of frames by the theorems of structural variation", International Journal for Numerical Methods in Engineering, Vol. 21 pp. 671-681 Moharrami, H. and Reyazi M. M., (2000) "Analysis of structures including Compression-only and Tension-only members", proceeding of European Congress on Computational Methods in Applied Science and Engineering (Eccomas 2000), Barcelona, Spain.

Owen, D. R. J. and Hinton, E., (1980). "Finite elements in plasticity – theory and practise" Pineridge press Swansea.

Riyazi Mazloumi, M., (1999)" Analysis of structures including truss elements with limited axial strengths", M. Sc. thesis, Tarbiat Modarres University, Tehran, IR Iran.

Saka, M. P., (1997). "The theorems of Structural variation for Grillage systems" Innovation in Computer Methods for Civil and Structural Engineering, Edinburgh, Civil Comp Press, pp. 101 111 Saka, M. P., (1991). "Finite element application of the theorems of structural variation", Journal of Computers and Structures, Vol. 41, No.3, pp. 519-530

Saka, M. P., and Celik, T., (1985) "Nonlinear analysis of space trusses by the theorems of structural variation", Proceeding of the second international conference on Civil and Struc. Eng. Comp. CIVIL – COMP 85, Vol. 2, pp. 153-158