Inelastic analysis of sections subjected to axial force and bending moment

Pergamon

Compurers & Svu~rurs~ Vol. 59. No. I, pp 13-19. 1996 Published by Elsevier Saence Ltd Printed tn Great Eirilam 0045.7949196 %15.00 + 0.00

00457949(95)00253-7

INELASTIC

ANALYSIS OF SECTIONS FORCE AND BENDING

SUBJECTED MOMENT

TO AXIAL

M. J. Terrot and S. A. Hamoushf tDepartment iNorth

of Civil Engineering,

Carolina

Kuwait

A&T State University,

University,

Greensboro,

Kuwait NC, U.S.A.

(Received 30 September 1994) Abstract-This paper describes the development of a numerical model used for the inelastic analysis of sections subjected to combined axial force and bending moment. The objective of the model is to calculate the strain distribution across the section given the applied loads and the mechanical properties of the material. The stress-strain relation is assumed to be bi-linear elastic-plastic in tension and compression. Exact solutions for six modes of strain in pure elastic, pure plastic and elastic-plastic in compression and tension are developed. A Newton-Raphson concept implemented in the iteration technique is used for

determining the strain values C, at the top and cJ at the bottom surfaces of the section. Although this analysis is for a rectangular section, this method is well suited to analyze members with more complicated cross-sections,

such as circular

or triangular.

1. INTRODUCTION

characteristics of the section starting from a knowledge of the applied loads. Since the solution of the nonlinear system is of an exact nature, unlike traditional methods, a much simpler and accurate computer model can be envisaged.

The nonlinear behavior of structural materials has been covered by many researches over the past years. The plastic analysis of sections subjected to combined axial and transverse loading is well established in the literature. The most commonly used solution methods are the finite element approach, the equivalent systems method and the analysis based on elastic-perfectly plastic assumption. The finite element approach is widely used in literature to model the inelastic behavior of sections [l-4]. The material nonlinearity could be accommodated by performing a through-thickness integration. This is achieved by subdividing the section into a number of layers of a predefined finite thickness. Knowing the strain values at each layer, the total force and moment could then be integrated over the section. The solution of finite element problems usually requires large capacity programming. The equivalent systems method was established by Ferdis ef al. [5-71. This method is used to analyze a nonprismatic elastic section. The method was further improved to include the non-elastic behavior of the material [8,9]. This method, however, is not suitable for analyzing sections subjected to combined axial force and bending moment. The elastic-perfectly plastic assumptions of the material behavior was used in Ref. [IO] to analyze sections subjected to combined axial force and bending moment. In the present model, an exact solution is developed relating the strains at the extreme sides of the section to the applied axial load and bending moment. The load-strains relation is developed for six different modes of strain. A numerical iterative procedure is then recruited to determine the deformation

Z.,MATHEMATICAL

FORMULATION

The problem is to establish a mathematical relation between the applied loads and the deformation state of the section, which is defined by the strain distribution. Different relations have to be used depending on whether the section is in a state of pure elastic, pure plastic or elastic-plastic behavior in tension or compression. These relations, coupled with an iterative approach, are then used to determine the strain values at the extreme sides of the section. In order to achieve the coupling, the constitutive stress-strain relation has to be first established. 2.1. Constitutiue

model

A bi-linear stress-strain relation is assumed in the model in the tension and compression zones (see Fig. 1) [l I]. An elastic modulus of elasticity, E,, controls the behavior of the material up to a yield strain eY. Beyond the yield strain t, a plastic modulus of elasticity, EP, is used to relate stress and strain. 2.2. Relation between loads and strains Assuming a linear strain distribution, the deformational state of the section is defined by the strain values c, at the top and Loat the bottom surfaces. For each set of values of C, and Q there exists a unique set of values for axial compression P and bending moment M. The mathematical relation between the values of P and M and the strains t, and c2 varies for 13

M. J. Terro and S. A. Hamoush

14

Stress

G

Fig. I. Constitutive stress-strain each mode of strain. The exact values of P and M are derived from the following relations: (1) M=

OydA,

(2)

s where y is the vertical distance from dA to the centroid of the section and G is the applied stress at each dA. Bending around one axis is only considered, Using the above relations for P and M and based on the assumed constitutive stress-strain model, the exact values of the axial load and bending moment corresponding to the modes of strain are presented in Appendix A. 2.3. Modes of strain Depending on the values of E, and E* in the above stress strain model, six different modes of strain are indicated. Three modes are in the compression and three are in the compression-tension states. These modes of strain are represented as follows: Mode 1. Compression-elastic state cy > 6, > t2 > 0. Both strain values are in compression and below the yield strain. The section behaves elastically. See eqns (Al) and (A2). Mode 2. Compression-elastic/plastic state t, > .zy> Ed> 0. E, and c2 are still in compression. E, is in the plastic region while t2 is still below the yield state. Part of the section will behave plastically. See eqns (A3) and (A4). Mode 3. Compression-plastic state 6, > c2 z ~~> 0. In this mode L, and t2 reach the plastic state in compression. The whole section will behave plastically. See eqns (A5) and (A6).

relation.

Mode 4. Compression/tension-elastic state t, z E, > 0 > cl > -t, E, is still in the compression zone and t2 in the tension zone. Both strain values are below yield. The section behaves elastically. See eqns (A7) and (As). Mode E, > cy > yield in tension behaves

5. Compression/tension-elastic/plastic state 0 > cl > -t,.. In this mode c, goes beyond compression, while cz is still in the elastic zone. Part of the section in compression plastically. See eqns (A9) and (AIO).

Mode 6. Compression/tension-elastic~plustic. stute tl > ty > 0 > - ty > ez. E, and t? go beyond yield in the compression and tension zones, respectively. Two extreme parts of the section will perform plastic behavior (see Fig. 2). See eqns (All) and (A12). Figure 2 illustrates the various modes of strain discussed above. It should be noted that t, is always assumed to be greater than t?. Therefore, the elastic/plastic compression where t2 > t). > t, > 0 and compression/tension where cy > t, > 0 > -6). > t? are derived directly from the above modes in which t, and c2 will be determined by changing the sign of the applied moment. The true values of the strains will be obtained by switching the calculated strains. 3. NUMERICAL

APPROACH

Using the exact mathematical formulation of P and M discussed above, the objective of the model is to find values of E, and E? starting from a knowledge of the loads applied. Since the strains are not known a priori, the problem is to determine which mode of strain describes the behavior of the section under the applied loads. Therefore, a numerical approach is developed as described below.

Sections subjected to axial force and bending moment

‘E”

se

In

E,

section

‘&Y

EY

K

15 &Y

E,

I

E2

(a) - Strain Mode 1

(b) - Strain Mode 2

section

(c) -

(e) -

&y

(d) - Strain Mode 4

Strain Mode 3

Ed

section

El

section

E,

Strain Mode 5

(f) -

Strain Mode 6

Fig. 2. Elastic and inelastic modes of strain.

An initial set of values for L, and cZ is first assumed, based on the assumption that the section behaves elastically under pure compression or pure bending. The section would be then in either of mode 1 with pure axial load P or mode 4 with pure bending moment M. The choice between the two modes 1 and 4 is governed by the relative values of initially applied loads P and M. Based on the assumed values of E, and Q, calculated values of P and M are obtained. The difference between the applied and calculated values of axial load and bending moment AP and AM are derived. An iterative procedure is then employed to minimize AP and AM.

3.1. Iterative method An iteration procedure based on the concept of the Newton-Raphson technique is employed to minimize the values of AP and AM below a preset tolerance. A good convergence rate was achieved using this method which is explained below. A horizontal translation of the strain diagram is employed to minimize AP, as illustrated in Fig. 3. Once the value of AP is minimized, a rotation in the strain diagram is then performed to achieve convergence in M. This method of minimizing AM is illustrated in Fig. 4. This iterative procedure is then repeated until both AP and AM are simultaneously minimized below a

M. J. Terro and S. A. Hamoush

16

section

section

(a) - Decrease

(b) - Increase

P Fig

3. Correction

oft,

and t2 to minimize

P

AP. AE section

section

+

&2 t-

E2

(a) - Decrease

(b) - Increase

M Fig. 4. Correction

of c, and t2 to minimize

preset tolerance value. A tolerance value of 1% is used in this model. At every iteration step in this approach, the mode of strain is calculated for the newly corrected values of c, and t2 and the appropriate mathematical relation is used in calculating P and M. To improve the convergence of the iterative system, some kind of limitation was required to make the transfer between modes smooth. Therefore, on initial comparison of the ratio of bending moment to the applied load, it was concluded that if this ratio is less than H/6 (where H is the depth of the section) the whole section will be in a state of compression and modes 1, 2 and 3 will be applicable only. Otherwise, if this ratio is greater than H/6, the bottom strain t2 will be negative (in tension) and modes 4, 5 and 6 will be applicable. Since the variables t, and ez are moving targets between modes, the above limitation helped to improve convergence. Table

Axial load x lOa(N) 0.15 0.221 0.40 0.15 0.15 0.15

-+El

3.2. Computer

M

AM.

model

The method explained above is implemented in a computer program called “CURVATURE”. The applied axial load P, and bending moment M, together with the mechanical material and geometric properties of the section constitute the input to the program. The output data of the program consists of the strain values t, and E* with the value of the radius of curvature R. A flow chart of the program CURVATURE is presented in Fig. 5.

4. EXAMPLE

RUN

An example run of the model is presented in this section. The axial load and eccentricity values were chosen in such a way that all six modes of strain are I

Bending moment x lo8 (N.mm)

Eccentricity (mm)

6, x 10-2

t2 x 10-z

Mode

2.1 3.9 1.2 7.5 14.2 25.5

14.2 11.6 3.0 49.8 94.9 170.0

0.159 0.369 0.675 0.123 0.303 0.574

0.088 0.135 0.618 -0.122 -0.103 -0.372

1 2 3 4 5 6

Sections

subjected

to axial force and bending

moment

17

covered. The error level between computed and given values of axial load and bending moment has been preset Material

(

STtRT

to 1%. properties

)

E, = I.8 x 1O’MPa;

I

Applied

E, = 5.7 x IO4 MPa;

INPUT Pa ‘and Ma

t, = 0.28 Section Assume initial

c

x 10-2.

properties

I and ~2 B=

150mm;

H = 300 mm.

5. DISCUSSION AND CONCLUSIONS

The developed method is based on exact mathematical solutions, unlike most of the traditional methods used to analyze nonlinear materials. The error is preset to 1% and could be changed to tighter or slacker values. Therefore, it has an improved level of accuracy over the traditional methods. Since it employs exact questions, the model is simpler to use, easier to model and faster to run. The method presented above leads to further development, to include the nonlinear analysis of sections with complex geometry in which an integration through the thickness will be required. Such types of analyses are in progress by the authors. The method could be recruited in future research work to investigate the behavior of elements of structure exhibiting nonlinear material behavior which would be usually difficult to analyze using traditional methods. An example of such research would be non-prismatic structures, such as tapered beams or columns subjected to a combined axial load with initial eccentricity with lateral distributed load, simpler, easier to model, faster to run, opens room to sections, beams, more research, non-prismatic columns.

Calculate Corrcspond~ng MC

I

I

Calcularr Corresponding PC

I

I

AP=Pa-PC I

Acknowledgements-The authors would like to express their gratitude for the unfailing support of Kuwait University. The authors are also thankful to the Computer Center of Kuwait University for the hardware and software facilities which were made available for research purposes.

NO AP < Tolemce

Rfor

Fig. 5. Flow chart

elk

of model.

c?

REFERENCES

N. El-Mezani. C. Balkaya and E. Citipitiogly, Analysis of frames with non-prismatic members. J. srrucr. Div. ASCE 117, (1990). R. Funk and K. T. Wang, Stiffness of non-prismatic member. J. sfrucf. Dir. ASCE 114, 484494 (1988). Resende and Doyle, Non-prismatic and effective threedimensional non-prismatic beam finite element. Comput. Srruct. 14, 71-77 (1981). I. Mumuni, A finite element model for the analysis and optimal design of beams and plates with variable flexural rigidity. Thesis presented to Vanderbilt University at Nashville, TN. In partial fullfillment of the requirement for the degree of Doctor of Philosophy (1983).

M. J. Terro

18

and S. A. Hamoush

5. D. G. Ferdis and A. Kozma, Solution of deflection of variable thickness plates by the method of equivalent systems. J. Ind. Marh. 12, (1962). 6. D. G. Ferdis and E. Zobel, Equtvalent system for the deflection of variable stiffness members. J SIYUC~.Die. ASCE (1958). Theor)‘, 7. D. J. Ferdis and E. Zobel, Transverse Vdvorion Application of Equivalent Sysrems. Roland Press. New York (1961). 8. D. G. Ferdis and M. E. Keene. Elastic and inelastic analysis of non-prismatic members. J. ~fr~ct. Div. AXE 116, ( 1990). Equivalent systems 9. D. G. Ferdis and R. Taneja, for inelastic analysts of prismatic and non-prismatic members. J. SI~UCI. Div. AXE 117, (1991) 10. H. G. Allen and P. S. Bulson, Background IO Buckling. McGraw-Hill, Maidenhead (1980). Il. S. Timoshenko, Strength of Materials, Parr I. Elementary Theory and Proh1rm.r. 3rd Edn. Robert E. Krieger Publishing, Huntington.

Modr 4

where A,=-

‘I _ , _‘! i 62>

Mode 5

h h fzE,bhz + L2. Ep”+ __2

APPENDIX A

2

(A9)

VALUES OF THE AXIAL LOAD AND BENDING MOMENT CORRESPONDING TO THE MODES OF STRAIN

Mode

1 PEt

2

Ef,h+!?++C’) E,hh

(Al)

__

t

2

where

Mode 2 P = c,h,E e b + (c,

E&h

2

+c,h,E,b

+(y!h&,

(A3)

Mode 6 P =;[2c,E,h,Zb +r,h&b($

- h, -;),

(A41

+ E,(q - c?)h,,b - 2c,E,h&

I M = j c&h,,b(h

where h,

+ 1;1,(<: + c,)h,,bj

(All)

- h,:)

=@I --ylh (6, - t2)

h, = h -

h,

+t,E,bh,,(;-h,,-+)

Mode 3

+t,E,hz,b(h P=c,E,bh+czE,bh+-

M=(f,-t,)%.

(6,

-62)

2 E bh*

E

bh

p

- hTz)

(A5)

(A6)

6412)

Sections subjected to axial force and bending moment where

19

In the above equations: 6, is the strain at top face of steel, face of steel. E, is the elastic modulus of elasticity, E, is the plastic modulus of elasticity, cY is the yield strain of steel, h,, is the depth of elastic compression, h,* is the depth of plastic compression, h,,is the depth of plastic tension, h, is the depth of compression and h2 is the depth of tension. It should be noted that 6, and t2 are positive in compression and negative in tension. t2 is the strain at bottom

h2 = h - h,.