Buckling analysis of the internal bars in square pyramid space grid

Architectural Engineering May 2013, Volume 1, Issue 1, PP.1-5

Buckling Analysis of the Internal Bars in Square Pyramid Space Grid Wenjie Niu College of Mechanics and Engineering Department, Liaoning Technical University, Fuxin Liaoning 123000, China Email: nwj1982@sohu.com

Abstract The correct determination of the internal load of the internal bars in square pyramid space grid structure is essential while the buckling analysis of the internal bars is more important. Buckling analysis of the internal bars in square pyramid space grid structure when the joints are hinges was done. Internal loads of the internal bars in square pyramid space grid structure were analyzed when the joints are fixed connections, which was first order analysis. Buckling analysis which is second order analysis would be done in the future.

Keywords: Square Pyramid Space Grid Structure; Hinge Joints; Fixed Connection Joints; First Order Analysis; Second Order Analysis

1 INTRODUCTION Spatial grid structure is a spatial structure made by several member bars arranged in grid form and connected with joints. The merits of spatial grid structure are light weight, great rigidity, and good performance of shock resistance.

Spatial grid structure could be used as roofs of gym, cinema, exhibition hall and so on. The demerits of spatial grid structure are that there are too many bars connected at the joint and the production and installation processes are more complex than plane structures [1].

One of spatial grid structures is plate type spatial grid structure [1]. And one of plate type spatial grid structures is square pyramid system space grid [2]. These square pyramid space grid structures could be seen as in Fig.1 and Fig.2.

FIG.1 SQUARE PYRAMID SPACE GRID PUT STRAIGHT [3]

FIG.2 SQUARE PYRAMID SPACE GRID ROOF [3]

The correct determination of the internal load of the internal bars in square pyramid space grid structure is essential while the buckling analysis of the internal bars is more important [4].

-1http://www.ivypub.org/ae

2 BUCKLING ANALYSIS OF THE INTERNAL BARS IN SQUARE PYRAMID SPACE GRID STRUCTURE WHEN THE JOINTS ARE HINGES

FIG.3 QUADRANGULAR PYRAMID

For the quadrangular pyramids in Fig.1 and 2, if the resulting force of dead load and live load on roof plate ABCD is vertically applied on the centroid of plate ABCD and its value is P (see left part of Fig.3). If the bar AO, BO, CO and DO is axisymmetric, the inverted view for the structure is shown in the right part of Fig.3. Take node O as research object, the external force is PAO , PBO , PCO , PDO and reaction force P. Because AO, BO, CO and DO is axisymmetric and P is vertically applied on the centroid of plate ABCD, then [4]

P P P P AO

BO

CO

(1)

DO

These forces are in equilibrium in horizontal plane. To ensure the equilibrium of these forces in vertical direction [5],

sin   P  sin   P  sin   P  sin   P  P AO

BO

CO

DO

(2)

Substitute (1) into (2), then

P

PAO  PBO  PCO  PDO 

4 sin 

(3)

If the joints at A, B, C, D and O are hinges in Fig.3, the critical buckling load (elastic stability limit) for bar AO ( BO, CO or DO) is given by Euler's formula as [6, 7]

Fcr 

EI 

2

(4)

2

L

Where EI is the flexural rigidity of bar AO, BO, CO or DO. L is the length of bar AO, BO, CO or DO in Fig.3.

With (3) and (4), if bar AO, BO, CO or DO is under critical buckling load (elastic stability limit), the resulting force P of dead load and live load on roof plate ABCD (see left part of Fig.3) is:

Pcr 

4 sin   EI 

2

2

L

(5)

So to secure the bar AO, BO, CO or DO not buckling, the resulting force P of dead load and live load on roof plate ABCD (see left part of Fig.3) could not exceed Pcr.

3 INTERNAL LOAD DETERMINATION OF THE INTERNAL BARS IN SQUARE PYRAMID SPACE GRID STRUCTURE WHEN THE JOINTS ARE FIXED CONNECTIONS If the joints at A, B, C, D and O are fixed connections as in the left part of the Fig.4, assume the joint O vertical displacement is LOE as in the right part of the Fig.4. The compression deformation along bar OD is LOF as in the right part of the Fig.4. The deflection at point O of bar OD is LOG as in the right part of the Fig.4.

-2http://www.ivypub.org/ae

FIG.4 INTERNAL LOAD DETERMINATION WHEN THE JOINTS ARE FIXED CONNECTIONS

The compression load Rc (presented in the right part of the Fig.5) on joint O because of bar OD compression deformation LOF is [8]:

Rc 

LOF EA L

(6)

Where E is the modulus of elasticity of the bar and A is the cross-sectional area of the bar.

The shear load Rs (presented in the right part of the Fig.5) on joint O because of deflection LOG is [8]:

Rs 

12 EI LOG L3

(7)

The bending moment M (presented in the right part of the Fig.5) on joint O because of deflection LOG is [8]:

M

6 EI LOG L2

(8)

FIG.5 LOADS ON JOINT O

The resulting vertical load on joint O provided by bar OD against the vertical displacement of joint O in Fig.5 is:

ROD  Rc * sin   Rs * cos 

(9)

Combining (6), (7) and (9) yields

ROD 

LOF 12 EI EA * sin   3 LOG * cos  L L

(10)

The resulting resistant vertical load RO on joint O against the vertical displacement of joint O provided by bar AO, BO, CO or DO in Fig.5 is: -3http://www.ivypub.org/ae

RO  4* ROD

(11)

Joint O is in equilibrium by the external load P and resulting resistant force RO .

RO  P

(12)

Combining (10), (11) and (12) yields

P LOF 12 EI  EA * sin   3 LOG * cos  4 L L

(13)

See Fig.4, it is obvious that

LOF LOE L cos   OG LOE sin  

(14) (15)

Combining (13), (14) and (15) yields

P4

LOE 48EI EA * sin 2   3 LOE * cos 2  L L

(16)

Eq. (16) could be transformed as:

LOE 

P 4 48 * EI * EA * sin 2   * cos 2  3 L L

LOF P * A * sin  EA  48 * I L 4 * A * sin 2   2 * cos 2  L 3 * I * P * cos  Rs  2 A * L * sin 2   12 * I * cos 2  3 * I  P * cos  M 24 * I 2 * A * L * sin 2   * cos 2  L

Rc 

(17)

(18)

(19) (20)

The above analysis for square pyramid space grid structure when the joints are fixed connections is first order analysis or typical structural analysis. Internal loads of the internal bars in square pyramid space grid structure were analyzed when the joints are fixed connections, and the results were presented in (18), (19) and (20).

Typical structural analysis is on the un-deflected structure within the material mechanics assumption [9, 10] while buckling analysis is not the case. When a beam as in Fig.5 buckles because of axial compressive force, moment and lateral force, the problem is more complex than the original traditional buckling problem [11, 10]. The above analysis done was just first order elastic analysis [12]. The critical buckling load (elastic stability limit) for above square pyramid space grid structure may be analyzed in the future [13].

4 CONCLUSIONS Buckling analysis of the internal bars in square pyramid space grid structure when the joints are hinges was done. To secure the internal bar AO, BO, CO or DO not buckling, the resulting force P of dead load and live load on roof plate 2 ABCD (see left part of Fig.3) can not exceed Pcr which is 4 sin  EI .

L2

Internal loads of the internal bars in square pyramid space grid structure were analyzed when the joints are fixed connections. The results were presented in (18), (19) and (20). The analysis was first order analysis. Buckling analysis which is second order analysis would be done in the future.

ACKNOWLEDGMENT The author is supported by the Scientific Research Starting Funds at Liaoning Technical University (No.11-415).

-4http://www.ivypub.org/ae

The financial help is greatly appreciated.

REFERENCES [1]

Baidu Encyclopedias.“Spatial grid structure.”Accessed on http://baike.baidu.com/view/52100.htm, October 24th, 2012 (in Chinese)

[2]

Baidu Encyclopedias.“Square pyramid system space grid.”Accessed on http://baike.baidu.com/view/1164175.htm, October 24th, 2012 (in Chinese)

[3]

Shanghai Jiguang Polytechnic College. “Types of space grid.” Accessed on http://jpkc2.shjgxy.com/jzsgjs/upload/2011/7/8/ 87220827183.swf, October 24th, 2012.(in Chinese)

[4]

Wen-jie Niu. “Internal Load Determination of Two Spatial Structures: a Simple Roof Truss and a Quadrangular Pyramid.”Advanced Materials Research, 2012,Vols. 378-379, pp: 349-352

[5]

Stephen P. Timoshenko, Donovan Harold Young.Theory of Structures.New York: Mcgraw-hill Inc., 1968

[6]

Efunda.“Euler's Formula.”Accessed on http://www.efunda.com/formulae/solid_mechanics/columns/columns.cfm, October 24th, 2012

[7]

Qin-shan. Fan and Xin Cai. Material mechanics.Beijing:Tsinghua Univ. Press,2006 (in Chinese)

[8]

Lian-kun Li. Structural mechanics.Higher education press, 2004 (in Chinese).

[9]

Qin-shan Fan. Engineering mechanics.China Machine Press, pp.232-247, 2004.(in Chinese)

[10] T.B. Quimby.“A Beginner's Guide to Structural Engineering. ” http://www.bgstructuralengineering.com [11] Alexander Chajes. Principles of Structural Stability Theory. Waveland Pr Inc, pp.138-165,1993 [12] Baidu Encyclopedias. “First order elastic analysis.” Accessed on http://baike.baidu.com/view/8345448.htm, October 25th, 2012.(in Chinese) [13] Hui-fa CHEN. Theories of beam-columns. Beijing: Science Press,1997

AUTHORS Wenjie Niu was born at Luohe City in He'nan (or Henan) Province of China in March 2nd, 1982. He was educated from 1999 to 2009 in Department of Geotechnical and Underground Engineering in Tongji University.

He obtained his bachelor's degree and doctor's degree in geological engineering from Tongji University in Shanghai of China.

He completed his post doctoral research in Hydraulic Structure and Water Environment Research Institute from 2009 to 2011 in Zhejiang University. He works in Liaoning Technical University as an associate professor on Mechanics and Engineering from 2011 to now. A few of his papers were published in Rock and Soil Mechanics (in Chinese), Engineering Mechanics (in Chinese) or Mathematical Problems in Engineering. His previous research interests include slope safety analysis and design, structural mechanics and concrete dam stress analysis. Current interests include experimental mechanics, strength theory of steel and concrete, buckling analysis of beam and pillar, and wind turbine foundation analysis and so on.

Associate Prof. Niu was a member of ARMA and ISRM in 2010. He was a reviewer of Hydrogeology Journal, Journal of Chongqing University, Frontiers of Structural and Civil Engineering, and Proceedings of the ICE - Bridge Engineering.

-5http://www.ivypub.org/ae