An elastoplastic analysis of a beam element is presented … elastoplastic analysis of 1-D...

On elastoplastic analysis of 1-D structural member

subj ect to combined bending and torsion

M. Moestopo

Structural Mechanics Laboratory, Inter-University Center for

Engineering Sciences, Bandung Institute of Technology,

Jl Ganesha 15, Bandung 40132, Indonesia

Abstract

A new model of an elastoplastic rectangular beam element is presented. Themodel is used in an elastoplastic analysis of the beam due to combinedbending and torsional moment by introducing a plastic-depth factor. Using theplastic node method, the yielding process of a section can be included in theanalysis without modeling the beam as a plane or a solid structural member.Therefore, the computational cost on an elastoplastic analysis is reduced.Examples shows that the present analysis gives a more realistic andconservative collapse load.

I. Introduction

In an elastic-plastic analysis, the yielding in a beam element is usually assumedto be controlled only by bending. However, for member that also carriestorsional moment such as in grillage structures, the interaction betweenbending and torsion at yielding element should be considered.

The plastiilcation in a beam member is often assumed to lump at a plastichinge section which has a zero length while the material is assumed to beelastic perfectly-plastic. The yielding initiates on edges of a section when thebending and the torsion satisfy the fully-plastic condition, i.e. the analysisinvolves only the elastic and the fully-plastic section. A collapse load obtainedby these assumptions is expected to be higher than the load obtained byconsidering the plastification process of the section.

Transactions on the Built Environment vol 17, © 1996 WIT Press, www.witpress.com, ISSN 1743-3509

3 1 4 Engineering Integrity

An elastoplastic analysis of a beam element is presented by the author whichincludes the interaction between bending and torsion during the plastification atthe plastic hinge. To accomodate the plastification of layers at the yieldingsection, an elastoplastic beam model is introduced so that the beam does notneed any mesh or layer refinement.

2. Plasticity and Plastic Node Method

Yielding is assumed at any point of a beam element due to combined normaland shear stresses when the von Mises yield condition is satisfied :

where ciy is the normal yield stress of the material due to axial tension. Thiscondition at a fully-plastic beam section is satisfied when the bending moment,M, and torsional moment, T, at that section satisfy the yield interaction [1] :

(2)

where Mp is the fully-plastic bending moment of the section in pure bendingand Tp is the fully-plastic torsional moment of the section in pure torsion. Theyield function,/ can be written as

-( M\* (T\*/ = W + ur ~* (3),,

where / = 0 refers to a yield condition.

The Plastic Node Method was introduced by Ueda [2] for an elastic-plasticanalysis of structure by assuming the plasticity is lumped as a fully-plasticsection at beam-ends (i.e. plastic node).

The elastic force-displacement relation can be written as :

k u = r (4)

where u is the displacement vector of the beam element, k is the beamstiffness matrix, and r is the force vector. The displacement increment, 5u, canbe written as


Engineering Integrity 315

where suhsricpts / and / refer to two beam-ends, superscript e and /; refer to theclastic and the plastic part of the displacement. For the associated flow rule theplastic displacement increment is expressed as

M - W £ (6)

a/

d f~d~r

d fd r

(7)

and [X ] is the scalar factor of proportionality matrix or plastic multipliermatrix. The increment of forces can be expressed as

^ » fa] IX M|5<1 Tk, k,l|6u,-5u,"|

^̂ -|5r,|-[k, kJ{8u';P|_k,, kjsu.-fiuff

and

5r,

5r.

(8)

(9)

where the zero value of 5// or 5^ is assumed if node / or/ is plastic.For a plastic node of a standard beam element where r = [Q% M^. '\\

and T/ can be written in the form of

0 2vy Z, (10)

where subscript 6 refers to the plastic node / or/.The two equations. Equation 8 and Equation 9, can be used to construct the

force-displacement relation for the 1-D beam element,

wliere superscript cp indicates the elastic-plastic stiffness matrix.


316 Engineering Integrity

As illustrations, let consider two cases of plasticity of a beam element.Case 1 : Node / is fully-plastic, node7' is still elastic.In this case, 8// = 0 and X, = 0, and the elastic-plastic stiffness matrix becomes

, , , , ,k,, - kYYk

k -

k, -

where :

or

where :

(',1

Y.r k '

k'" = k - n.

Q. =u,

(12)

(13)

(H)

(15)

The matrix Q, describes the stiffness matrix decrease due to the fully-plasticsection (plastic node) which is assumed at end-node /.

Case 2 : Node / is still elastic, node j is fully-plastic.In this case, S/J = 0 and A, = 0. The elastic-plastic stiffness matrix becomes

where := k - Q,

and

(16)

(17)

(18)

3. Elastoplastic Analysis

The plastiilcation of a section due to combined bending and torsion is differentfrom the plastification in pure bending or in pure torsion individually. Thecomplexity in the analysis may arise if the analysis must include the yielding ofthe material along the length, the width, and the thickness of the beamelement. Also, the ratio of bending to torsional moment will vary during theplastification.



An elastoplastic section that models the plastification of a rectangular beamsection is introduced as shown in Figure 1. This model accomodates theyielding of the section via a plastic depth factor, a, which describes how far theplastification has occured from both edges of the section toward the center ofthe section, due to combined bending and torsion. At an elastoplastic section,the normal stress, a* and the shear stress, i* acting on the yielding regionmust satisfy the yield condition in Equation 1.

A. =A

M

AC

Ap

1 ~1 & 2

M

i«l

(a) (b)

Figure 1 : A Rectangular beam section model.a. Elastic Section b. Elastoplastic section

The bending moment, M, acting on the section can be written as asummation of MI , Mi and M-,, as shown in Figure 2 :

M, = (/7-a6)(a

A/, = £ />(/7-a/7)'(7-ot)a*

(19)

(20)

(21)

Although the model does not satisfy the compatibility on the yieldingsection, nevertheless the bending moment contribution from normal stress nearthe neutral axis is relatively small compared with the one from normal stressacting near the bottom and top layer of the section.



M,

(h-ab)

i -a)b

(h-ab)

figure 2 : Bending moment, M, acting on the elastoplastic section model.M = M, +M2 + Mr,

The lorsional moment, T, can be written as a summation of Ti.T], and T.i :

b\~ a- ~~r •

4- 2 |(/, - a /,) . ^ . ^ . T *

4- 2J(/7 -a/)). ̂ . ̂ . T *

= (a'6'(/, + /, _ ja/?)T*

(22)


Engineering Integrity 3 1 9

7^ = 2 (A - a 6) (A - a A) (^) % * (23)

7, = c (/? - a A) (6 - a /;)' T * (24)

where T, and 1^ are equal to two times the volume of the solid bodies shownin Figure 3. and c\ can be approximate by the Roarks's formula [3] :

(25)

Figure 3 : Torsional moment, T, acting on the elastoplastic section model



The bending and the torsional moment at a yielding section can be written as

M = M, + M, + M,= / , ** (26)

6 ' '"

r = 7; + 7; + 7;(27)

= /r /? T* • 8,

where :r / , ^\i

(28)

a + c,(/-4 + i %' {/ + (/-"«) 77

(29)

When the edges begin to yield, a = 0, 8,,i = 1 and 8t = c$, while for a fully-plastic section, a = 1, 8^ = 1.5 and 8, = ( 3 - b/h)/6. These values agree withthe analytical values for clastic and fully plastic section.

Substituting the values of a* and T* in Equation 26 and Equation 27 intoEquation 1 will give a new yield criteria for an elastoplastic section :

\ 2I 4-1 ̂ -̂ ~ I (X-l = / _Q,

The plastic-depth factor, a, can be computed numerically for any appliedbending and torsional moment on an elastoplastic rectangular section, by usingEquation 38 and Equation 39.

The stiffness matrix of a yielding member decreases. To obtain aconservative collapse load, the values of moment of inertia of yieldingmembers are approximated by /^ = co I and J^ = co J, where co = (7-a){ Thus,the elastic-plastic stiffness matrix of the beam element, k^, is then computedas

k''" = k - ( 1 - o)) Q (31)

Once a section becomes fully-plastic, co = 0, and Equation 31 becomesEquation 14 or Equation 16.



4. Some Examples

a. A right angle bent structure due to a concentrated vertical load shown inFigure 4 was analysed by Ueda [2], Heyman[4], and Hodge[5] to obtain thecollapse load by assuming the elastic and the fully-plastic section. Thebeam properties are £ = 3(10)* psi, G = 1.2(10)* psi, o>,= 20,000 psi, h = h= 0.75 in., L = 20 in. Previous analysis gives a collapse load P = 499 Ibswhen the vertical deflection at node B is about 1.0 in. The presentelastoplastic analysis gives P = 480 Ibs with the load-displacement curveshown in Figure 5. The first yielding occurs at node D when P = 220 Ibs.In present analysis the variation of bending and torsional moment is non-linear with a smooth transition from elastic to elastoplastic state (Figure 6).After the edges of the section yields at P = 220 Ibs, the rate of change of thebending moment decreases while the torsional moment increases faster thanin elastic state. Figure 7 shows the change of (M/M,, )* + (T/Tp f and theplastic-depth factor, a , of the section in node D.

b. An inelastic lateral buckling of intersecting connected beams shows inFigure 8 was analysed using the present analysis. The beam properties are :£=3 (10)* psi, G= 1.16(10)* psi, GJ,= 20,000 psi, fr = 0.125 in, h = 5 in., I,= 20 in, /,? = 10 in. All supports are fixed.The stability analysis by Morchi [6] gives an elastic buckling load P = 5,020Ibs, while the virtual work method gives an upper bound collapse load P =4.688 Ibs. By including the stability matrix in the analysis, the presentelastoplastic analysis gives an inelastic buckling load P = 3,925 Ibs as shownin Figure 9.

5. Conclusion

A model of plastification of a rectangular beam element has been presented.The model can be used to perform an elastoplastic analysis of structure withbeam member where the yielding of the beam section is governed not only bythe bending but also by the torsion at the beam-ends. The inelastic buckling canalso be detected using the analysis.

The present analysis gives a more realistic and conservative collapse load.Since the beam element is analysed as a 1-D element and not by a planeelement or a solid element to include the yielding proccess at a beam section,the analysis could save considerably large computational cost.



Figure 4 : A Right angle bent structure under a vertical load

(x 10 Ibs)

Elastic Fully-plastic Analysis

Elastoplastic Analysis

1 ' 2

Deflection at Node B (inch)

Figure 5 : Load-displacement relation of structure in Figure 4



Elastic

10 20 30 40

(x 10 Ibs)

Figure 6 : Bending and torsional moments at Node D in Figure 4.Dash lines corresponds to previous analysis.

Figure 7 : The variation of plastic depth factor, a, on yielding section at node D



IZ, Ll L1

L2

Figure 8 : A Rigidly connected cross-beams.

100 Ibs)

50P = 4.625 Ibs

30-

20 J

Inelastic BucklingP = 3.925 Ibs

1st yieldingat P = 2.325 Ibs

Elastic Fully-plastic AnalysisElastoplastic AnalysisElastoplastic and Stability Analysis

0.0 0.2 0.4 0.6 0.8 1.0

Center Deflection (inch)

Figure 9 : Load-displacement relation of structure in Figure 8



References

1. Chakrabarty. J., Theory of Plasticity, McGraw-Hill Inc., New York, 1987.2. Ueda, Y. and Yao, T., The Plastic Node Method : A new method of plastic

analysis. Computer Methods in Applied Mechanics and Engineering, 1982.34, 1089-1104.

3. Young.W.C., Roark's Formulas for Stress and Strain, McGraw-Hill Inc.,New York, 6ed., 1989.

4. Hey man. .1., The limit design of space frames, Journal of AppliedMechanics. 1951, 18, 157-162.

5 Hodge, P.G., Plastic Analysis of Structures, McGraw-Hill Inc, New York,1959.

6. Morchi. S.P. & Lovell, E.G. Lateral Buckling of intersecting connectedbeams. Journal of the Engineering Mechanics Division, 1978, 104, EM2,367-381.

7. Moestopo. M.. On Ultimate Strength Analysis of Grillages Structures. PhD.Thesis, University of Wisconsin-Madison, 1994.


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