Advanced analysis and design of spatial structures

ELSEVIER

J. Construct. Steel Res. Vol. 42, No. 1, pp. 2 1 4 8 , 1997 © 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain PII: S0143-974X(97)00~5-9 0143-974X/97 $17.00 + 0.00

Advanced Analysis and Design of Spatial Structures J. Y. Richard Liew, N. M. Punniyakotty & N. E. Shanmugam National University of Singapore, Department of Civil Engineering, 10 Kent Ridge Crescent, Singapore 119260 (Received 4 March 1996; revised version received 28 November 1996; accepted 16 December 1996)

ABSTRACT Modern limit-state design codes are based on limits of structural resistance. To determine the 'true' ultimate load-carrying capacity of spatial structures, an advanced analysis method which considers the interaction of actual behaviour of individual members with that of the structure is required. In the present work, a large-displacement inelastic analysis technique has been adopted to compute the maximum strength of spatial structures considering both member and structure instability. The actual behaviour of individual members in a spatial structure is depicted in the form of an inelastic strut model considering member initial imperfections as 'enlarged' out-ofstraightness. The maximum strength of the strut is computed based on a member with 'equivalent out-of-straightness' so as to achieve the specification's strength for an axially loaded column. The results obtained by the strut model are shown to agree well with those determined using plastic-zone analysis. The nonlinear equilibrium equations resulting from geometrical and material nonlinearities are solved using an incremental-iterative numerical scheme based on generalised displacement control method. The effectiveness of the proposed advanced analysis over the conventional analysis~design approach is demonstrated by application to several space truss problems. The design implications associated with the use of the advanced analysis are discussed. © 1997 Elsevier Science Ltd.

NOTATION

A A~ D

E,E~

{F~_,}

--

Area of cross-section of a member Effective area of cross-section of a tension member Outer diameter of a tubular member Modulus of elasticity and plasticity Vector of internal element forces summed at each node of 21

22

GSP GSP i I [ K i-1]

[Kj-1] L, L o

P Pe

Pmax

Py Pul

{Pj} {P} Y

{R}-l} t

W

Z 8o 6 {6U~}

(6u3} 8Al Ae

J. Y. R. Liew et al.

the structure up to the (j-1)th iteration of ith load increment Generalised stiffness parameter Generalised stiffness parameter at ith load increment Second moment of area Tangent stiffness matrix of the structure at the beginning of the load increment i Tangent stiffness matrix formed at the beginning of the jth iteration based on the known element details at ( j - 1 ) t h iteration of ith load increment Chord length and arc length of a member Full plastic moment capacity Reduced plastic moment capacity in the presence of axial force P Applied axial force Euler buckling load capacity of a strut member Axial load capacity of a strut member as per design specifications Axial load capacity at full yielding of cross-section Axial load at which unloading begins in a strut member Vector of total external nodal loads on the structure at jth iteration of ith load increment Reference load vector on the structure Radius of gyration Vector of unbalanced forces during ( j - 1)th iteration of ith load increment Wall thickness of a tubular member Lateral deflection at a distance x from one end of the strut member Plastic section modulus Member initial out-of-straightness magnitude at mid-span Member mid-span deflection Iterative displacement vector at the structure level in jth iteration of ith load increment Iterative tangential displacement vector at the structure level in jth iteration of ith load increment Iterative residual displacement vector at the structure level in jth iteration of ith load increment Initial iterative load parameter specified as input Iterative load parameter for the jth iteration of ith load increment Total elastic shortening in a strut

Advanced analysis and design of spatial structures a

a

age, a max

Ap Ay

{APi} {APi} PE O'y

23

Member axial shortening due to compression Member axial shortening due to change of geometry in the elastic and plastic range Member axial shortening before elastic unloading Total member axial shortening in the plastic range Member elongation at yield load Incremental load vector on the structure during the ith load increment Incremental displacement vector of the structure during the ith load increment Specified energy tolerance Design yield strength of member material

1 INTRODUCTION Spatial structures having rigidly connected joints are more complicated to analyse because each joint consists of at least 6 degrees of freedom, and the members are subjected to combined actions of axial force, bending, torsion and shears. However, rigidly jointed structures have the advantage of less member density because they are stiffer and require smaller member sizes to satisfy the design limit states. Experimental studies have shown that actual spatial structures generally fall between the rigid jointed and pin-jointed analytical models, even though the actual connections may appear to have been pinned or rigid. Actual joint rotational stiffness is difficult to quantify for design, because it depends on many factors such as installation sequences, lack of fit, types of nodes and their configurations, member and system imperfections, etc. If the joint stiffness cannot be quantified for analysis, the assumption of pin-jointed members should be used because it would lead to a conservative design and safe estimate of deflection. For structures with higher rigidity against instability, such as double layer grids and deep domes, it is a common practice to assume that the structure behaviour is linearly elastic. Elastic first-order analysis of such structures is straightforward, commercial programs are available and approximate methods can be applied for manual computations. However, to ensure member stability, the forces obtained from the analysis must be checked against the specification of member equations. These equations consider stability of an individual member, and they implicitly include member imperfection effects, such as residual stresses and initial out-of-straightness. However, there is no theoretical reason to suppose that the capacity determined through member capacity equations for the most critical member in a general, especially a slender spatial structure, represents the limit load of the structure. To compute the true limiting strength

24

]. Y. R. Liew et al.

of such structures, an advanced analysis, which can capture the interaction effect of local buckling of members and overall buckling of structure, is required. Advanced analysis refers to an analysis for all relevant load combinations of which specification limit states checks are satisfied implicitly and thus explicit verification of design limit states are not required. Work on advanced analysis of two-dimensional semi-rigid frames has been reported recently [1,2]. Educational software for advanced analysis and design is also available in book form [3,4]. Advanced analysis is useful for slender spatial structures which are susceptible to geometric instability and member buckling. This is particularly true for single-layer reticulated shallow domes, or for domes with small rise-tospan ratio, in which the structures may exhibit several types of instability associated with member buckling, local or dimple buckling at a joint, and interaction of local and overall instability. Substantial discrepancies have been observed between the structural behaviour as predicted by linearly elastic analysis and the actual behaviour [5,6]. Murtha-Smith [7] presented the chronological development of various analytical strut models, and highlighted the limitations of their use in actual design practice. The basic problem of these analytical models is that they do not satisfy the code requirement for member capacity checks, which therefore deters one from adopting them for use in actual design. In the present work, an inelastic large-displacement advanced analysis, which is capable of capturing individual member strength and overall buckling of structure, is presented. The strut model is based on a physical analogy that one strut element is used to represent each truss member in a spatial structure, and the strut capacity is made to conform with the specification's member capacity equation. A numerical scheme using generalised displacement control algorithm has been adopted for solving the nonlinear equations. This numerical scheme has been shown to be capable of handling many kinds of instability problems associated with slender spatial structures.

2 ELEMENT MODELS

2.1 General description The present model is based on a physical analogy that one strut element is used to represent each truss member in a spatial structure, and the strut capacity is made to conform with the specification's member capacity equation. The main objective is to minimise computing time. Further, one element per member approach is oriented towards conventional design philosophy since typical elements in a numerical model correspond to components to be checked in

Advanced analysis and design of spatial structures

25

conventional design. However, compared to conventional numerical analysis based on updated Lagrangian formulation of so-called co-rotational formulation, the element model implies a coarse element representation because each element between the nodal points is modelled by only one element. Therefore, it is necessary to take into account geometric nonlinearities at the local element level in order to capture the effect of member buckling. The present formulation handles large displacements on local and global levels. On the local level, large deflection is incorporated when a plastic hinge is formed at the mid-span of the element, in which the element is assumed to be rigid plastic; whereas the global effects are taken in by updating nodal point coordinates. Thus, a total Lagrangian formulation is implemented on the element level. However, the program does not imply a complete total Lagrangian formulation since the element reference axes are updated based on the deformed geometry at each load step. 2.2 Strut model

In the present analytical model, the member initial out-of-straightness is assumed in the form of a half-sine wave. The load-displacement curve of a completely elastic strut with a sinusoidal initial bow is combined with that of a plastic-hinge curve to form an elastic-perfectly plastic strut model. This concept was originally proposed by Papadrakakis [8] and was further extended by Maheeb [9] for struts with various types of cross-section for planar bracing systems. The present study extends Maheeb's approach to three-dimensional systems. The behaviour of a strut under the action of axial load shows three distinct characteristics, as shown in Fig. 1. They are discussed in the following subsections. 2.2.1 Elastic pre-buckling curve The first stage of the load-displacement curve (curve AB in Fig. 1) is associated with the compressive loading of an initially imperfect column that buckled at point B. During this stage of loading, the total elastic axial shortening, Ae, due to an applied axial force P consists of two components: Ae = A a + A~.

(1)

The first component, Aa, is due to the axial deformation. The second component, Ag, is associated with the change of geometry due to bow in the strut. With reference to Fig. 2, these two components may be written as: Aa -

PLo EA

(2)

J. Y. R. Liew et al.

26

Axial Load, P

Py

7

.ineor~///.~astic

~a~ p .x,C"--

Curve(exact}

---~-'~8~----_/ElasticLoadingCurve(exact) 7

PE

-p - ~ ~ ~,

t

Prnox Put

-//,Ao

2 Af /~b Ab

Amox

Axial Shortening,A

Fig. 1. Axial load-displacement curve for a strut member.

L

V

L

L_.. ~ _ _ ~

1~

_1

I~°

P

Fig. 2. Imperfect strut in the elastic range.

Ag = Lo - L

(3)

where L is the member's chord length, and L0 is the member's arc length given by:

Lo=

1+

(4)

in which w is the lateral deflection at a distance x from one end of the member as shown in Fig. 2. For a strut with an initial out-of-straightness in the form of a half-sine wave and maximum deflection 30 at the mid-span, the total lateral displacement including the second-order displacement due to the axial force effect may be written as [ 10]:

Advanced analysis and design of spatial structures w(x)_6o

1-

['rrx] p sin ~ -

27 (51

Pc

where Pe is the Euler buckling load of the strut. The member's arc length Lo can be computed by substituting eqn (5) into eqn (4) and perform the integration using a parabolic approximation, which can be shown to compare well with the numerical integration [9]. With chord length equal to L and total mid-span lateral deflection equal to 8, eqn (4) may be approximated as: Lo=L

(6)

l + 3\Loj J

in which the mid-span deflection, 6, can be computed by letting x = L/2 in eqn (5) as: 6 = w(x = L/2) -

(7)

p. 1-

Pe

Substituting eqn (6) into eqn (3),

(1)

(8)

2_(287

1 + 3\Lo/

From eqn (1), the total elastic axial shortening can be computed as follows:

Ae=Lo ~ +

(9)

2]28 : " 1+ 5

Substituting 6 from eqn (7) into eqn (9), the axial force-shortening relationship of a strut in the elastic range can be computed as:

Ae=Lo ~ + 1 -

(8

3o

1 + 3 Lo(1 - P/Pe

)"2

(10)

28

J. Y. R. Liew et al.

2.2.2 Plastic p o s t - b u c k l i n g curve

The curve BC in Fig. 1 is characterised by a decreasing axial load accompanied by column shortening after the axial load P reaches its maximum strength, Pmax • The shortening is primarily due to lateral deflection of the member, which facilitates the through-section and along-length plastification of the member at mid-span caused by the P - 8 effect. Once the strut begins to buckle, yielding develops rapidly, and the axial force decreases with the increase in axial shortening and the latter is related to the lateral deflection. The present model is developed by assuming that a plastic hinge is formed at the member mid-span as shown in Fig. 3. Neglecting the flexural deformation due to the elastic bending of two halves in comparison with those due to the plastic rotation at the mid-span, the lateral deflection at mid-span, 8, can be written as: (11)

8 - m _ MpJMpZ P P/Py A

where Mp = full plastic moment capacity, Mpc = reduced plastic moment capacity in the presence of axial force P, Py = axial load capacity at full yield, A = cross-sectional area and Z = plastic section modulus of the cross-section. From Fig. 3, the axial shortening due to geometrical change, Apg, can be derived as:

\G/]"

(12)

By adding the axial deformation Aa with Ag, the total axial shortening, in the plastic range is given as:

I_

P

[.

I-

•

P

Fig. 3. Strut in the perfectly plastic range.

Ap,

Advanced analysis and design of spatial structures

A p = A a + A g= Lo EA + 1 -

1 - \Loo]

"

29

(13)

Substituting 3 from eqn (11) into eqn (13), the axial force-shortening relationship of a strut in the plastic range can be computed as:

,p

j1

(14)

2.2.3 Elastic post-buckling unloading curve

Curve CD in Fig. 1 is due to elastic unloading of the strut at the post-collapse range. Unloading occurs when the strut member is stretched due to the change in sign of the incremental force. If unloading occurs in the elastic range, the same elastic pre-buckling curve is followed; otherwise, the post-buckling unloading curve is developed based on the total axial shortening, Amax, and the axial load, Pu~, at the unloading point as shown in Fig. 1. Letting Ae = mmax and P = Pu~, the equivalent member out-of-straightness magnitude 60 can be computed from eqn (10). When the 60 value is known, the elastic postbuckling unloading curve can be generated using eqn (10). 2.2.4 Plastic hinge yield surface

A plastic hinge is inserted at the mid-span of the member when the crosssectional forces, computed from a second-order analysis, reach the section plastic strength. Once a plastic hinge is formed, the cross-section forces are assumed to move on the plastic strength surface. The expressions for Mpc/Mp in eqn (11) are dependent on the type of cross-sections. Plastic strength equations, which represent the relationship between the axial force and moment on the yield surface, can be found in [11] for typical doubly symmetric crosssections as given below: For rectangular section: Mpc/Mp = 1.0

-

(p/py)2.

(15)

For circular tubular section: Mpc/Mp = cos[( qr/2)(P/Py)].

For I- or H-section bending about the weak axis:

(16)

J. Y. R. Liew et al.

30

Mpc/Mp = 1.0 for 0 --< P --< 0.4Py

(17)

MpdMp = 1.1911.0 - (p/py)2] for 0.4Py < P _< Py. For I- or H-section bending about the strong axis:

MpdMp = 1.0 for 0 --< P -< 0.15Py

MpdMp=

1.1811.0 -

P/Py]for

(18)

0.15Py < P --< Py.

2.2.5 Design implementation The load-carrying capacity of an axially load strut can be made to conform with the code requirement for the design of columns. The code formulae for axially compressed columns account implicitly for the effects of member imperfections such as initial out-of-straightness and residual stresses. Therefore, the strut model, with its ultimate load computed from the design code, would satisfy the code's intention for column design including the member imperfection effects. One major advantage of such approach is that explicit modelling of member imperfections can be avoided in the analysis. In the present formulation, the multiple column curves given in BS5950 (1990) [12] have been adopted to compute the maximum strength of strut members depending on the types of cross-sections. The procedure to obtain the axial force-deformation relation of a strut is given below: (1) From the known material, cross-sectional properties and length of a strut member, the axial load capacity, Pmax, is computed based on BS5950 (1990) [12]. (2) Using the plastic post-buckling formulas eqn (14), compute the total axial shortening Ap corresponding to P = Pmax. (3) At P = Pmax the elastic pre-buckling curve intersects the plastic postbuckling curve, and the total axial elastic shortening Ae from eqn (10) is equal to Ap from eqn (14). The corresponding member initial outof-straightness magnitude, 30, can be back calculated from eqn (10) by letting P = Pmax and me = m p . (4) Given the value of 60, the elastic pre-buckling curve can be generated for various values of P up to Pmax using eqn (10). (5) When the strut is deformed beyond Ap which is shown as Ab in Fig. 1, the strut is in the plastic post-buckling range. Equation (14) should be used to compute the axial force-shortening relationship. (6) When the incremental axial displacement shows a reverse sign, unloading from the strut curve is expected. The unloading curve from the plastic post-buckling curve is assumed to follow the elastic pre-

Advanced analysis and design of spatial structures

31

buckling curve with member imperfection magnitude computed based on initial shortening value, Af, as shown in Fig. 1. The procedure has been described in section 2.2.3.

2.2.6 Verification of the strut model Sugimoto and Chen [13] analysed the load--deformation behaviour of tubular members using the finite segment method based on generalised cyclic stressstrain relationships in the form of moment-axial force-curvature and axial force-moment-axial strain relations and an automatic load control technique. Chan and Kitipornchai [14] also developed theoretical axial load-shortening curves for pin-ended tubular struts based on the fundamental stress-strain relationship of the material and compared with those of Sugimoto and Chen [13]. Sugimoto and Chen carried out their analyses on three pin-ended columns with maximum initial out-of-straightness magnitude of 0.001 L at the mid-span. The column member was divided into 13 finite segments along half of the length. Tubular sections were used with outer diameter D = 114.3 mm, wall thickness t = 2.4 mm, elastic modulus E = 2 × 105 MPa, and design strength O-y = 248 MPa. The column slenderness ratios used in the study are 80, 120 and 160. Figure 4 shows the comparison between these results [13] and those obtained by the present strut model based on the same maximum strengths (Pmax) obtained by Sugimoto and Chen [13]. It is seen from the figure that there is a general agreement between the load-displacement results except in 1.0

P b

0.0

~

L

q P

.

- Sugirnoto& Chert(1985) PresentModel

.. 0.6 P

Py O.l, 0.2 0.0

0.000

0.002 Axiat

0.00/, Strtzin,AlL

Fig. 4. Comparison of strut curves with refined model.

0.006

32

J. Y. R. Liew et al.

the post-buckling range. This is because the proposed strut model is based on member with 'enlarged' initial imperfection and elastic-perfectly plastic hinge behaviour whereas in the Sugimoto and Chen model the cross-sectional behaviour is elasto-plastic with gradual plastification and spread-of-plasticity within a finite length. The 'enlarged' imperfection approach adopted in the proposed model makes the load-displacement curve more flexible towards the inelastic region. However, the inaccuracy due to such approximation should be acceptable for design purposes. 2.3 Tie model

The axial load-displacement relationship of a tie member is assumed to be linearly-elastic with strain-hardening when the applied tensile force exceeds the yield load Py as shown in Fig. 5. Assuming axial shortening as positive, the governing equations may be written as:

P-

E IAI for IA[ -< lay

p-

EAe ~- my -- ~ ( I A I

- Imyl) for

(19)

IAI>IAyl

(20)

where Py : Aetry is yield load, my is the axial elongation due to the tensile force equal to Py, Ae is the effective area under tension, O-y is the material yield strength, E is the modulus of elasticity, and Ep = E/IO00 is the strainhardening parameter. Axial Load, P

ell

/ LIE.

Ay

Axial elongation. A

Fig. 5. Axial load-displacement curve for a tie member.

Advanced analysis and design of spatial structures

33

3 SOLUTION ALGORITHM

3.1 Incremental-iterative equilibrium equation Based on the updated Lagrangian formulation, the incremental equilibrium equation of the structure can be written as: [K i- I]{AU i} = { A p i}

(21)

where [Ki-l] is the tangent stiffness matrix of the structure obtained at the beginning of the load increment i and {AUi} is the incremental displacement vector of the structure in response to a load increment of {APi} during the ith load increment. Equation (21) can be solved by a simple incremental technique. The major deficiency of this technique is the risk for drift off from the exact solution path and the difficulty in bypassing the limit points on the loaddisplacement path. Correction for this uncertainty is taken care of by including equilibrium iterations in the load vector at every load increment by means of the generalised displacement control method.

3.2 Generalised displacement control method The generalised displacement control (GDC) method originally proposed by Yang and Shieh [15] is selected to solve the incremental equilibrium equation given by eqn (21) considering its numerical stability near all types of limit points. With the superscript i denoting the current load increment step and the subscript j denoting the current iteration number, eqn (21) may be rewritten as: (22) where [Kj-d is the tangent stiffness matrix formed at the beginning of the jth iteration based on the known element details at the (j-1)th iteration, {rU)} is the iterative displacement vector obtained for the jth iteration, {Pj} is the total external nodal loads applied on the structure at jth iteration, and {~_ ~} is the internal element forces summed at each node of the structure up to the (j-1)th iteration--all during the ith load increment. For the case of proportional loading with a reference load vector {P} applied through a scalar iterative load parameter o-Aj, eqn (22) can be further modified as: (23) where {Rj_ ~} denotes a vector of unbalanced forces given as:

34

J. Y. R. Liew et al.

{R~_,} : {Pj_~} - { ~ _ 1 } .

(24)

The iterative displacement vector {6U~} in eqn (23) can be decomposed into two parts as:

{6v~} = 6~{60j} + {au-j}

(25)

where {80}} and {6U-}} are, respectively, the tangential and residual iterative displacement vectors expressed as:

{60~}: [K~_,]-,{p}

(26)

{6U3} [K~_I] {gj--1}. -1

i

=

(27)

A generalised stiffness parameter (GSP) is introduced to compute the iterative load parameter until the convergence of incremental equilibrium equation is achieved. The GSP i for any load increment, i, is defined as the ratio of the norm of the first iterative displacement vector of the first load increment step to those at the current load increment step:

{601F{6Ol}

GSP~= {60~- 1}T{68]}"

(28)

For the first load increment, the GSP 1 value is equal to one. The first iterative load parameter of the first load increment is equal to the input value, 6)tl and for the subsequent load increments it can be computed using the known GSP i value of that particular load increment as:

6A~ = • 6AIIGSPq ''z.

(29)

The iterative load parameter, 6)t} for all the subsequent iterations (j > 1) is computed using the following expression:

(30) As the GDC method is based on the bounded characteristics of load parameter and displacement increments, it is able to bypass both the snap-through and snap-back limit points without causing any numerical instability.

Advanced analysis and design of spatial structures

35

3.3 Convergence criteria For each load increment, the equilibrium equation is solved by iterations till the unbalanced force vector {Rj._~} becomes negligible. This is indirectly achieved by satisfying the following energy criteria: i T

i

l{SU~} {R~-I}I

(31)

in which ~ is a user-specified tolerance. It can be a value between 1 x 10 -6 and 1 x 10 -1°. To terminate the iteration for non-converging and slowconverging systems, a maximum number of iterations per load step is also imposed.

4 NUMERICAL EXAMPLES

4.1 Two-bar truss system This numerical example is intended to demonstrate the effect of member imperfections on the limit load of the structure and to compare the limit load predicted by the advanced analysis with the conventional elastic analysis. A two-bar truss system, shown in Fig. 6, consists of square bars of 0.254 m x 0.254 m size with member slenderness ratio L/r of 150. The modulus of elasticity of material is E = 2.06 × 105 N/mm 2 and yield strength, O'y = 235 N / m m 2. The truss supports are restrained against translations. The structure is subjected to a concentrated downward load, P, applied at the crown joint resulting in compressive forces in both the members. The structure is analysed using, (1) large-displacement elastic analysis in which the axial force-shortening relationship of the members is assumed to

I

P -~.695

L i

10.977rn

_J_ 7-

10.977m

_J i

Fig. 6. Two-bar truss system.

rn

J. Y. R. Liew et al.

36 5000

l Elastic/ z r ~

t'6181

2500

0 A

0.000

E

0.005 0.010 0.015 0.020 Axial shortening (m)

0.025

Fig. 7. Axial load-shortening curves for two-bar truss members.

be linearly elastic as shown by the dashed line in Fig. 7, and (2) advanced analysis in which the axial force-shortening relationship of the members is assumed to be nonlinear as shown by the solid curve in Fig. 7. The actual nonlinear behaviour of these strut members is obtained using the strut model proposed earlier in section 2.2. The strut capacity, P . . . . is computed as 4618 kN using the column curve 'b' of BS5950 [12]. The load versus vertical displacement curves at the crown joint of the two-bar system obtained by the two analyses are shown in Fig. 8. For the advanced analysis, the load-shortening behaviour of the members corresponding to different stages of loading are labelled as A, B, C, D and E in Figs 7 and 8. The limit load predicted by the large-displacement analysis assuming linearly elastic member behaviour is 1313 kN, whereas the consideration of member nonlinearity by the advanced analysis yields a limit load of 445 kN. The advanced analysis shows that the structure attains its limit load at point B (Fig. 8) before full member capacity of 4618 kN corresponding to 1500

Elasticlarge

1000

displacement alysis

500

"~ ~

/

/

o. -SO0 -1000 -1500

80

o'.s

1'.o

Vertical displacement(m)

1.s

Fig. 8. Load-displacement curves for the crown joint of two-bar truss.

Advanced analysis and design of spatial structures

37

point C (Fig. 7) is reached. The failure of the two-bar system is due to the geometric instability, with the members attaining their maximum strength at the post-collapse range corresponding to point C in Fig. 7. Point D in Fig. 8 corresponds to the case where the crown joint of the truss is deflected up to the support level. The deflection beyond point D will lead to unloading of the member at the post-buckling range as indicated by curve DE in Fig. 7. If one adopts an elastic analysis and applies the conventional member capacity check approach to determine the limit load of the structure, the summation of vertical components of individual member capacity of 4618 kN works out to be 584 kN. As this value is well below the buckling load of 1313 kN given by the elastic large-displacement analysis, it would be treated as the limit load capacity of the truss. This way, the limit load of 584 kN obtained by conventional analysis and design overestimates by about 30% from the true limit load of 445 kN obtained by the advanced analysis. The performance of advanced analysis reveals that the consideration of member imperfections during the geometric nonlinear analysis triggers the structure to attain its unstable condition at an earlier stage. 4.2 S p a c e d o m e s y s t e m

This numerical example is presented to demonstrate the effect of external loading on the limit load of the structural system. Figure 9 shows a spatial truss system consisting of 24 members to form a star-shaped structure. The member cross-sectional properties are: A = 10 mm2; 1 = 41.7 mm4; E = 2.034 × 105 N/mm 2, O-y = 400 N/mm 2 and Ep = E/1000. All the supports are restrained against translations and the remaining nodes are free to translate in the space. The slenderness ratios (L/r) for members 1-6 and 13-24 are 123 and 155, respectively.

18V17

PLAN

t

62.16mm

• Supports ELEVATION

Fig. 9. Star truss system.

J. Y. R. Liew et al.

38

_

633

~oo j

o

X

g

1'o

1s

Vertical displocementat crown (ram) Fig. 10. Load-displacement curves for the crown joint of star truss under load case 1.

The space truss is analysed for two load cases, (1) a single downward concentrated load applied at the crown joint and (2) downward concentrated loads applied at all the seven unrestrained joints. For the advanced analysis, the axial load-shortening curves for the strut members are obtained based on the column curve 'b' of BS5950 [12] and the tension members are modelled as linearly elastic with strain-hardening defined by the plastic modulus Ep. The structure is analysed by both elastic large-displacement analysis and advanced analysis, and the relationships between the applied load, P, and the vertical displacement at the crown joint of the structure are shown in Figs l0 and I 1 for load cases 1 and 2, respectively. The axial load-shortening curves for the truss members are shown in Figs 12(a) and (b) for load case 1 and in Figs 13(a)-(c) for load case 2. The oneto-one correspondence between the behaviours of structure and members are labelled as A, B, C and D in Figs 10 and 12(a) and (b) for load case 1 and as A, B and C in Figs I I and 13(a)-(c) for load case 2. For the load case l, the ultimate load capacity of the structure predicted by the elastic large

2000. ~ 1500- : ~ 15/,3 "Loadedjoints . ~ " -~ 1000" ~ x-Etosticlarge / displocement -/ anolysis .~500- /~Advonced onolysis 'I: o

~ ~ ~ ~ lo Vertical displacementat crown (ram)

Fig. 11. Load-displacement curves for the crown joint of star truss under load case 2.

Advanced analysis and design of spatial structures 800.

39 ....

~7/.t,

1500 600. II

0

I

./L ~-members 9{-861~ -758} I to 6 /"C(-! 'I) ) -1500members~ / 7 to 12 "V i I "~ -3000/

a

200,

I

J

-LSO0

-0.8

-o'.L

o.o

o:~

Axid shortening {mm)

(Q)

i°

LO0. --_=

o.B

0 O0

B(157) C(136) 0(51)

~2

o'.~

0'.6

~ot shortening (ram)

{b)

Fig. 12. Axial load-shorteningcurves for star truss members. (a) 1-12; (b) 13-24 under load case 1. displacement analysis and advanced analysis are 633 and 372 N, respectively. The corresponding limit loads for load case 2 are 10,801 N (7 × 1543 N) and 1659 N (7 × 237 N), respectively. By observing the label B in Figs 10, 11, 12(a) and 13(c), it can be seen that the limit load of the truss corresponds to the geometric instability of the system and is not due to any member instability under load case 1 whereas it corresponds to the instability of supporting members 13-24 under load case 2. As per the conventional design, the vertical components of axial capacity of six top members 1-6 (1132 N) yields the limit load of the truss as 543 N under load case 1 as against 372 N obtained by advanced analysis making an overestimation of about 45%. Similarly, by considering the vertical components of 12 supporting members 13-24 (744 N), the limit load of the truss by conventional design works out to be 1756 N as against 1659 N obtained by advanced analysis making an overestimation of only 6%. This shows that if the limit load of the structure is governed by the instability of individual members and not due to the accelerated geometric instability of the system due to the inclusion of member imperfections, then the overestimation of limit load by conventional design is at an acceptable level. The analyses show that the structure can carry more load if the loads are distributed evenly to all the unrestrained nodes rather than concentrated at the crown node alone. The advanced analysis shows that the structure can carry about three times more load as load case 2 (1659 N) than load case 1 (372 N), whereas the conventional design (543 and 1756 N) shows about two times increase in the total applied load. The reason for the increase in load-carrying capacity due to the change of loading conditions from load case 1 to 2 can be explained as below: (1) Comparison of axial load-shortening curves for members 1-6 as shown in Fig. 12(a) with Fig. 13(a) indicates that the maximum compressive

J. Y. R. Liew et al.

40 1200.

==

.... J

800"

I I I

;B15031

•~

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(c) Fig. 13. Axial load-shortening curves for star truss members. (a) 1~6; (b) 7-12; (c) 13-24 under load case 2.

loads in members 1-6 reduced by more than 50% from load case 1 to 2, and the members exhibit unloading in the elastic range under load case 2 instead of attaining the maximum strength and loading into the post-buckling range as under load case 1. (2) Figures 12(a) and 13(b) show that load case 1 creates tensile force in members 7-12 whereas load case 2 creates only compressive force. (3) The supporting members (members 13-24) are subjected to compressive force under both the load cases. However, corresponding to the

Advanced analysis and design of spatial structures 21

,,, 22

41

23

18

Fig. 14. Deformed configuration at the maximum load under load case 1.

label B at which the structure reaches its limit load, the magnitude of axial force is only about 20% under load case 1 than under load case 2 as indicated in Figs 12(b) and 13(c). The above observations show that the concentrated load under load case 1 causes an instability failure due to a snap-through of the crown joint, as shown by the deformed structural configuration at the maximum load in Fig. 14. The distributed loads under load case 2 impede the snap-through behaviour of the crown joint and stress the horizontal members 7-12 in compression instead of tension causing all the 12 supporting members to be stressed to their individual axial capacity in comparison with load case 1 in which all the forces are concentrated mainly to the top six inclined members, 1-6. These combined effects lead to an enhancement of the limit load of the structure under load case 2. The maximum load under load case 2 is reached due to the instability of the supporting members 13-24 as shown by the deformed configuration in Fig. 15. Thus, the advanced analysis enables a better understanding of the role of each individual member in the structure and provides an accurate prediction of the member and system collapse behaviour. The use of elastic large-

S

15

-

Fig. 15. Deformed configuration at the maximum load under load case 2.

42

J. Y. R. Liew et al.

displacement analysis may sometimes lead to an unconservative estimate of the system strength and stability.

4.3 Twelve-bar space truss system This numerical example is presented to demonstrate the effect of member slenderness on the limit load of the structural system. A spatial truss system consisting of 12 tubular members is shown in Fig. 16. Tubular members of different diameters, namely CS-1 and CS-2, are used in the analysis to demonstrate the effect of member slenderness on the overall behaviour of the structure. The CS-1 section consists of an outer diameter D = 193.7 mm, wall thickness t = 10 mm, area of cross-section A = 5770 m m 2 and moment of inertia I = 2442 × 104 m m 4. The corresponding properties for CS-2 section are D = 168.3 mm, t --- 5 mm, A = 2570 m m 2 and I = 856 × 104 m m 4. The modulus of elasticity for material is E = 205 kN/mm 2 and yield strength, try = 275 N / m m 2. All the supports are restrained against translations and the remaining nodes are free to translate in the space. The slenderness ratios (L/r) of supporting members 1, 2, 11 and 12 are, respectively, 65 and 58 with CS1 and CS-2 member properties. The spatial truss is analysed with a load P applied at the central joint marked as 'A' and concentrated load 1.5P at the two end joints. The structure is analysed by elastic large-displacement analysis and advanced analysis, and the relationship between the applied load, P, at joint A and the vertical displacement, W, at joint A, and horizontal and vertical displacements U and W at joint B are presented in Figs 17(a)-(c) and 18(a)(c), respectively. For the advanced analysis, the axial load-shortening curves for the tubular strut members are obtained based on column curve 'a' of BS5950 [12]. Figure 19(a)-(d) shows the axial load-shortening curves of individual members at different stages of loading. Similar to previous numerical examples,

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• S~perts ," Loaded joints

PLAIt Fig. 16. Twelve-bar truss system.

Advanced analysis and design of spatial structures

43

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Fig. 17. Load-displacement curves for 12-bar truss by elastic large displacement analysis.

the one-to-one correspondence between the behaviours of structure and members are labelled as A, B and C in Figs 18(a)-(c)and 19(a)-(d). From Fig. 17, the limit loads predicted by the elastic large-displacement analysis for structures with sections CS-1 and CS-2 are 80,980 and 36,030 kN, respectively. The corresponding .values predicted by the advanced analysis are 1096 and 419 kN as indicated in Fig. 18(a)-(c).

44

J. Y. R. Liew et al. 1200

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1000

600 LO0 200 0

20 l,O 60 60 Oisplecement W at joint B (mm}

(c}

Fig. 18. Load-displacement curves for 12-bar truss by advanced analysis.

Advanced analysis shows that the limit load of the structure is governed by the instability of the supporting members as shown by the collapse configuration at maximum load in Fig. 20. The supporting members 1, 2, 11 and 12 reach their maximum strength as indicated by the label B in Figs 18(a)(c) and 17(a). The configuration of the structure is such that it is stiff enough to prevent any joint instability. Hence the limit loads predicted by the

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Fig. 19. Axial load-shortening curves for 12-bar truss members. (a) 1, 2, 11 and 12; (b) 6 and 7; (c) 3, 5, 8 and 10; (d) 4 and 9.

2

9 11

Fig. 20. Deformed configuration at the maximum load.

advanced analysis for structure with member properties CS-1 and CS-2 are in proportion to the axial capacity of the critical members. In the case of elastic large-displacement analysis, the structure limit loads are proportional to the member cross-sectional areas. By computing the limit loads based on the vertical component of axial capacities of governing members 1, 2, 11 and

46

J. Y. R. Liew et al.

12, the conventional design would yield P values as 1120 and 444 kN as against 1096 and 419 kN by advanced analysis. The overestimation is only 2%. This strengthens the previous observation made with the star truss example that if the limit load of the structure is governed by the instability of individual members and not due to the accelerated geometric instability of the system due to the inclusion of member imperfections, then the overestimation of limit load by conventional design is at the acceptable level.

5 CONCLUSIONS A powerful advanced analysis/design method based on inelastic large displacement analysis has been proposed to compute the maximum strength of spatial truss-structures without the risk of overestimating the maximum strength of the component members in the structure. The proposed strut model is made to conform with the code requirement for strut design. This approach removes, to a large extent, the limitation of the conventional analysis which requires member capacity checks and the modelling of member initial imperfections for representing individual member strength. From the studies carried out on the example truss systems, the following conclusions may be drawn: (1) The performance of advanced analysis with the elastic-rigid plastic modelling for strut members as per the design specifications predicts the 'true' limit loads of spatial truss structures. (2) The conventional structural design of performing elastic large-displacement analysis followed by individual member capacity checks may sometimes lead to over-prediction of the limit load capacity. If the limit load of the structure is governed by the instability of individual members and not due to the accelerated geometric instability of the system due to the inclusion of member imperfections, then the overestimation of limit load by conventional design is acceptable. (3) The member imperfections, member sizes and the type of load distribution on the structure have appreciable effects over the limit load capacity of the structure as revealed by the performance of advanced analysis on the three truss systems. (4) The advanced analysis, in addition to predicting the limit load of the structure, helps to identify the load sharing and force distribution mechanism of individual members forming the structural system. The identification of the critical members in the structure enables the engineer to redesign the system so as to enhance the structural resistance and performance.

Advanced analysis and design of spatial structures

47

The proposed method shows promise as an effective means of accounting for member initial imperfections without physically altering the frame geometry. The advanced analysis fulfils most of the design specification requirements and can be applied for the design of complex spatial structures in which system instability can be a critical factor for the assessment of the structural limit states.

ACKNOWLEDGEMENTS The investigation presented in this paper is part of the research programs on 'computer aided second-order inelastic analysis for frame design' being carfled out in the Department of Civil Engineering, National University of Singapore. The work is funded by research grants (RP920651) made available by the National University of Singapore.

REFERENCES 1. Liew, J. Y. R., White, D. W. and Chen, W. F., Limit-states design of semi-rigid frames using advanced analysis: Part 1: connection modelling and classification; Part 2: analysis and design. Journal of Constructional Steel Research, 1993, 26(1), 1-57. 2. Liew, J. Y. R. and Chen, W. F., Implications of using refined plastic hinge analysis for load and resistance factor design. Thin-Walled Structures, 1994, 20(1-4), 17-47. 3. Chert, W. F. and Toma, S., Advanced Analysis in Steel Frames: Theory, Software and Applications. CRC Press, Boca Raton, FL, 1994. 4. Chen, W. F., Goto, Y. and Liew, J. Y. R., Stability Design of Semi-rigid Frames. John Wiley and Sons, New York, 1996. 5. Liew, J. Y. R., Punniyakotty, N. M. and Shanmugam, N. E., Inelastic largedisplacement analysis of space structures. MINDEF-NUS Joint R and D Seminar, Defence Technology Group, Ministry of Defence, Singapore, January 1996, pp. 132-139. 6. Liew, J. Y. R., Shanmugam, N. E. and Punniyakotty, N. M., Advanced analysis of space structures considering member stability effects. Proceedings of the AsiaPacific Conference on Shell and Spatial Structures, ed. T. T. Lien. China Civil Engineering Society, Beijing, 1996, pp. 228-237. 7. Murtha-Smith, E., Compression member models for space trusses: Review. Journal of Structural Engineering, ASCE, 1994, 120(8), 2399-2407. 8. Papadrakakis, M., Inelastic post-buckling analysis of trusses. Journal of Structural Engineering, ASCE, 1983, 109(9), 2129-2147. 9. Maheeb, M. E. Abdel-Ghaffar, Post-failure analysis for steel structures. Ph.D. thesis, Purdue University, 1992. 10. Chen, W. F. and Lui, E. M., Structural Stability--Theory and Implementation. Elsevier, New York, 1986.

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11. Chen, W. F. and Sohal, I. S., Plastic Design and Second-order Analysis of Steel Frames. Springer-Verlag, New York, 1995. 12. BS5950, Structural use of steelwork in buildings. Part 1: Code of practice for design in simple and continuous construction: hot rolled section. British Standards Institution, London, 1990. 13. Sugimoto, H. and Chen, W. F., Inelastic post-buckling behaviour of strut members. Journal of Structural Engineering, ASCE, 1985, 111(9), 1965-1978. 14. Chan, S. L. and Kitipornchai, S., Inelastic post-buckling behaviour of tubular struts. Journal of Structural Engineering, ASCE, 1988, 114(5), 1091-1105. 15. Yang, Y. B. and Shieh, M. S., Solution method for nonlinear problems with multiple critical points. AIAA Journal, 1990, 28(12), 2110-2116.