First Order Analysis of Plane Frames with Semi-rigid Connections

S. C. Patodi J. M. ChauhanCivil Engineering Department Civil Engineering DepartmentParul Institute of Engineering and Technology Parul Institute of Engineering and TechnologyLimda, Vadodara – 391 760 Limda, Vadodara – 391 760

Abstract— In the present paper, instead of using fictitious members to represent elastic joints, the element stiffness matrix and fixed end forces are modified to include semi-rigidity of joints. In terms of usage, this is a simplified approach because it does not require estimation of the dimensions of the fictitious members according to joint stiffness. A first order analysis is attempted based on direct stiffness method to include the semi-rigid behavior of joints. To verify the success of the C++ implementation of the modified formulation, the results obtained for three plane frame examples are compared with those available in the literature; results are found in very close agreement.

Keywords- Linear-elastic analysis; fixity factors; semi-rigid joints; modified stiffness method.

I. INTRODUCTION

The analysis of plane frames is usually based on the assumption that all joints are rigid which implies that members meeting at a particular joint of the structure undergo the same amount of rotation, since there is no relative rotation of one member with respect to other. Also, at supports of a structure, it is generally assumed that either ideally fixed or ideally pinned condition exists. In reality, however, many rigid connections permit a certain amount of rotation to take place within the connection and most pinned connection offer a small amount of restraint against rotation. Thus, if a more accurate representation of such structures is desired, it is necessary to consider the connection as being flexible or semi-rigid.

According to relative translations and rotations that can occur at the joints of a structure, several types of semi-rigid connections are theoretically possible. Connections for shear, bending moment and axial thrust, in case of a plane frame, may all possess certain amount of flexibility, but the most important of these is the rotational type which transmits bending moment. This type of semi-rigid connection is considered in this paper in conjunction with the analysis of plane frames.

In order that the analysis of a structure with flexible connection be possible, it is necessary to have adequate data pertaining to the characteristics of structural connections [1]. Tests on many types of connection between a beam and a column indicate that some relationship exists between the

restraining angular displacement between the beam and column. This relationship can be represented graphically by curve giving the moment versus rotation properties of the joint. Up to some load, the connection behaves elastically. At higher loads, the joint begins to deform more and the angle of rotation increases rapidly [2]. However, at normal working loads, the connections may be assumed to be linearly elastic.

The behavior of semi-rigid connections may be simulated by means of equivalent “springs” or “fictitious members” [3].This method either estimates or calibrates the dimensions of the fictitious members according to joint stiffness. Another way to include the behavior of semi-rigid connections in the analysis is to modify the stiffness properties of the individual members having a semi-rigid connection at one or both ends [4]. In terms of usage, this is a simplified approach because it does not require estimation of the dimensions of the fictitious members according to joint stiffness.

The method presented here is based on the direct stiffness method which is a powerful analysis tool for framed structures of any type with a high degree of accuracy. Following are the specific objectives of this paper:

To include the semi-rigid behavior of joints in the matrix method of analysis of frames by modifying the element stiffness matrix and fixed end actions to include the rotational stiffness of the connections.

To develop a program in C++ for the analysis of frames to facilitate linear elastic first order analysis using the modified stiffness and fixed end actions.

To compare the frame performance predicted by using the semi-rigid joint analysis to frame performance predicted by traditional pinned and rigid connections.

II. FORMULATION

Consider a member with semi-rigid connections as shown in Fig. 1. This member has moments and reactions at ends as fixed ends but the values will be less than what it will be in case of complete fixation at ends. The ends behave exactly as springs but have a different stiffness values Sj and Sk which can be related to the fixity factors rj and rk at the two ends as

13-14 May 2011 B.V.M. Engineering College, V.V.Nagar,Gujarat,India

National Conference on Recent Trends in Engineering & Technology

rj = 1/ (1 + 3 EI / Sj L) and rk = 1/ (1 + 3 EI / Sk L)

where E = modulus of elasticity of member, I = moment of inertia and L is the length of the member. For a hinged connection, the fixity factor is zero; but for a rigid connection, the fixity factor is 1 i.e. 100 percent. Since the fixity factor varies from 0 to 100 percent, it is more convenient to use.

Figure 1. Member with displacement numbers

If M represents the moment at end of semi-rigid joint and the value of θ represents the rotational angle of member end, the applied forces on member increase the rotation angle and consequently the reaction at end will increase. For simplicity, generally the M- θ relationship is assumed linear but actually the stiffness of the spring which is M divided by θ follows nonlinear relationship. It depends upon the type of connection and its value is to be obtained by experimental studies [1].

For a linear model, only one parameter defining the stiffness of a connection is required. The value remains constant along the analysis procedure, without the requirement for updating the connection stiffness. This is the simplest connection model and the analysis based on this model is known as first order analysis. However, it is not accurate for large defections but can be used in linear analysis where the deflections are small.

Terms in the element stiffness matrix for semi rigid connection are obtained using the flexibility approach and concept of translation matrix [5]. To obtain the flexibility coefficients at the k-end, the member is fixed at the j-end. A unit force is applied in the x-direction at the k-end and flexibility coefficients F11, F21 and F31 at the k-end are obtained. Then a unit force is applied in the y-direction at the k-end and flexibility coefficients F12, F22 and F32 at the k-end are obtained. Finally, a unit moment is applied at the k-end to obtain the flexibility coefficients F13, F23 and F33. Inversion of the flexibility matrix yields the stiffness matrix [Skk] for the k-end. Then other sub matrices of SM are found using the translation of axes technique. Assuming fixity factors as rj andrk at the j and k ends of the ith member, the non zero terms in modified member stiffness matrix [SM] for a member with semi rigid connection corresponding to displacement numbers 1 to 6 are found as follows:

SM11 = SM44 = AE/LSM41 = SM14 = - AE/LSM22 = SM55 = 12EI (rj rk+ rj + rk)/L

3 (4 - rj rk)SM52 = SM25 = - 12EI (rj rk+ rj + rk)/L

3 (4 - rj rk)SM32 = SM23 = 6EI rj (2 + rk)/L

2 (4 - rj rk)

SM35 = SM53 = - 6EI rj (2 + rk)/L2 (4 - rj rk)

SM33 = SM66 = 4EI (3rj)/ L (4 - rj rk)SM36 = SM63 = 2EI (3rj rk)/ L (4 - rj rk)SM62 = SM26 = 6EI rk (2 + rj)/ L2(4 - rj rk)SM65 = SM56 = - 6EI rk (2 + rj)/ L2 (4 - rj rk)

Due to rotational semi-rigid connections with rigidity factors as rj and rk at the member ends, the axial reactions will not change and hence AML1 and AML4 will correspond to the horizontal reactions in a simply supported beam. However, the vertical reactions AML2 and AML5 are calculated by adding and subtracting respectively Cjk from the vertical reactions obtained at j- and k- ends in a simply supported beam where Cjk is given by

Cjk = (AML3 + AML6)/Li

Where the modified fixed end actions AML3 and AML6 are calculated with the help of the following relations:

AML3 = (4rj - rj rk) Mj/(4 - rj rk) + (2rj rk - 2rj ) Mk/(4 - rj rk)

AML6 = (2rj rk - 2rk) Mj/(4 - rj rk) + (4rk - rj rk ) Mk/(4 - rj rk)

III. NUMERICAL EXAMPLES

In the present work, a program is developed in C++ using modular approach. A number of functions are developed to calculate element stiffness matrix SM, rotation transformation matrix Rt, structural oriented stiffness matrix SMS, inversion of S matrix S-1, transpose of Rt matrix RtT and combined load vector Ac. These functions are called in the main function to generate overall stiffness matrix, to isolate Sff and Srf from rearranged stiffness matrix, to separate out Afc and Arc from Ac vector and finally to calculate the displacements, reactions and member end actions. A number of input and output files are created simultaneously to efficiently manage the execution of the program.

Figure 2 displays the first example of a simple plane frame having joint rigidity r at the ends of each member as shown in figure. The geometric data and member properties with intensity of loading are also given in the same figure. For the purpose of calculation of stiffness matrix EI is assumed as 1000 and AE is assumed as 106 in consistent units. Member end actions obtained, using the developed program, are compared in the Table I with the solution provided by Wang [6]. When the same problem is solved by considering all the joints as fully rigid, the answer for displacements at joint B and C are found as [0.7636 -0.0004 -0.0687 0.7636 -0.0004 0.048]T against the results obtained by considering the semi-rigid as [2.3245 -0.00041 -0.1194 2.3244 -0.000038 0.04359]T. Similarly, the support reactions at A and D joints are found as [-4.011 22.451 35.54 -7.99 25.549 53.525]T with fully rigid connections and [-5.592 22.753 36.965 -6.41 25.247 62.89]T with semi-rigid connections.

13-14 May 2011 B.V.M. Engineering College, V.V.Nagar,Gujarat,India

National Conference on Recent Trends in Engineering & Technology

Figure 2. Frame with all the joints as semi-rigid (Ex.1)

TABLE I. MEMBER END ACTIONS IN FRAME WITH SEMIRIGIDCONNECTION AT BEAM-COLUMN ENDS

Next, a portal frame example as shown in Fig. 3 is analyzed by considering semi-rigid connection at the beam ends only. Results obtained from two standard cases i.e. rigid and pinned are compared with semi-rigid connection. Two types of semi-rigid connections are considered, namely, Double Web Angle (DWA) and Top and Seat Double Web Angle (TSDWA). Results for these four cases are provide by Sekulovic and Salatic [3] considering AE = 701400 kN and EI = 3171 kN-m2 for columns (AB and CD) and AE = 903000 kN and EI = 5817 kN-m2 for beam (BC).

Figure 3. Frame with elastic connections at beam ends

Results obtained for horizontal displacement at node B and bending moment at support A are compared in Table II. In addition, normalized graphs are plotted for horizontal displacement at node B and bending moment at support A are shown in Fig. 4. These graphs are plotted as a function of the fixity factor (r changing from 0 to 1 with the interval of 0.1) by dividing these values with those obtained for the same frame with pinned connection i.e. r = 0.

TABLE II. FRAME WITH SEMIRIGID CONNECTIONAT BEAM ENDS

Horizontal Disp. of Node B (m)

BM at Node A(kN-m)Conn.

TypePresent Sekulovic Present Sekulovic

Rigid (r = 1)

0.00258 0.00257 2.524 2.524

TSDWA (r = 0.8)

0.00286 0.00287 2.636 2.639

DWA (r = 0.68)

0.00309 0.00310 2.724 2.728

Pinned (r = 0)

0.00757 0.00757 4.502 4.503

Figure 4. Influence of the connection flexibilty (Ex. 2)

Finally, a two storey frame (Ex. 3) with the dimensions and loading as shown in Fig. 5 is analyzed by considering linear elastic connection at the beam ends of the frame. Axial and flexural rigidities of the beams and columns are assumed as the same as specified in the earlier example. Table IIIexhibits the results obtained for the horizontal displacement at node C and bending moment at node A for the ideal (rigid and pinned) and semi-rigid (TSDWA and DWA) cases. Also, normalized results for horizontal displacement at C and bending moment at support A are plotted against the fixity factor in Fig. 6.

MA

DB

13-14 May 2011 B.V.M. Engineering College, V.V.Nagar,Gujarat,India

National Conference on Recent Trends in Engineering & Technology

Figure 5. Two storey frame example (Ex. 3)

TABLE III. TWO STOREY FRAME WITH ELASTIC CONNECTION AT BEAM ENDS

Horizontal Disp. of Node C (m)

Bending Moment at Node A (kN-m)Conn.

TypePresent Sekulovic Present Sekulovic

Rigid (r = 1)

0.002326 0.002335 1.171 1.171

TSDWA (r = 0.8)

0.002774 0.002785 1.237 1.239

DWA (r = 0.68)

0.003138 0.003151 1.290 1.292

Pinned(r = 0)

0.017658 0.017661 3.000 3.001

Figure 6. Influence of the connection flexibilty (Ex. 3)

IV. CONCLUSIONS

Results obtained for a plane frame with unequal rigidity factor, considering axial deformation, are found in close agreement with those given by Wang [6] by neglecting axial deformation which confirms the fact that for such simple frames consideration of axial deformation does not change the results noticeably.

A comparison of results obtained for example 1 for joint displacements and support reactions, considering the joints as semi-rigid and fully rigid, clearly indicates that the consideration of elastic connections make a noticeable difference in the analysis results. Further, inclusion of nonlinear behavior of semi-rigid connection will certainly make the significant difference in the analysis; particularly at higher loads.

For examples 2 and 3 results obtained by using modified stiffness and fixed end actions are compared with the results provided by Sekulovic and Salatic [3] by considering semi-rigid connections as rotational springs. Results are found within 0.5 %. Also, the normalized graphs plotted for joint displacement and support moment by varying the rigidity factor from 0 to 1 clearly indicates the influence of connection flexibility on the analysis results which is also evident from the results provided in Tables 2 and 3.

REFERENCES

[1] L. R. Lima, S. A. Andrade, P. C. Vellaso and L. S. Silva, “Experimental and mechanical models for predicting the behavior of minor axis beam-to-column semi-rigid joints”, International Journal of Mechanical Sciences, Vol. 44, 2002, pp. 1047-1065.

[2] S. L. Chan and P. P. Chui, Nonlinear Static and Cyclic Analysis of Steel Frames with Semi-rigid Connections, Elsevier, Oxford, 2000.

[3] M. Sekulovic and R. Salatic, “Nonlinear analysis of frames with elastic connections”, International Journal of Computers and Structures, Vol. 79, No. 11, 2001, pp 1097-1107.

[4] W. Weaver and J. M. Gere, Matrix Analysis of Framed Structures, CBS Publishers and Distributors, Delhi, 1986.

[5] D. I. Masse and J. J. Salinas, “Analysis of timber trusses using semi-rigid joints”, Canadian Journal of Agricultural Engineering, Vol. 30, 1988, pp 111-124.

[6] C. K. Wang, Indeterminate Structural Analysis, McGraw-Hill International Book Co., New Delhi, 1985.

MA

DC

13-14 May 2011 B.V.M. Engineering College, V.V.Nagar,Gujarat,India

National Conference on Recent Trends in Engineering & Technology