Design o Rectanguar Openings in Unbonded Post-Tensioned Precast Concrete Was Apri 21 Michae Aen and Yahya C. Kurama Report #NDSE-1-1 Structura Engineering Research Report Department o Civi Engineering and Geoogica Sciences University o Notre Dame Notre Dame, Indiana
Design o Rectanguar Openings in Unbonded Post-Tensioned Precast Concrete Was Apri 21 Report #NDSE-1-1 by Michae Aen Former Graduate Research Assistant and Yahya C. Kurama Assistant Proessor Structura Engineering Research Report Department o Civi Engineering and Geoogica Sciences University o Notre Dame Notre Dame, Indiana
ABSTRACT Recent research has shown that unbonded post-tensioned precast was can be used as primary gravity and atera oad resisting systems in seismic regions. This report investigates the use o rectanguar openings in the was to accommodate architectura, mechanica, and/or saety requirements. The openings can cause arge tensie stresses, and thus, cracking in the wa panes. Under atera oads, cracking in the panes can aso occur due to gap opening aong the horizonta joints between the panes and at the base. Bonded mid stee reinorcement is needed in the wa panes to imit the size o the cracks. The report describes an anaytica investigation on the behavior and design o was with openings or two oading stages: (1) vertica oads due to gravity and post-tensioning; and (2) combined vertica and atera oads. For each oading stage, critica regions in the wa panes where reinorcement is needed are identiied and design approaches are proposed to determine the amount o the required pane reinorcement. The eect o opening ength, opening height, wa ength, and initia stress in the was due to post-tensioning and gravity on the behavior and design o the was is investigated. This report may be downoaded rom http://www.nd.edu/~concrete/
TABLE OF CONTENTS LIST OF TABLES... viii LIST OF FIGURES... x LIST OF SYMBOLS... xvi ACKNOWLEDGMENTS... xxiv CHAPTER 1 INTRODUCTION... 1 1.1 Overview... 1 1.2 Unbonded Post-Tensioned Precast Concrete Was... 2 1.3 Was with Openings... 3 1.4 Objectives... 5 1.5 Scope and Approach... 5 1.6 Organization o Report... 7 CHAPTER 2 BACKGROUND... 8 2.1 Unbonded Post-Tensioned Precast Concrete Was... 8 2.1.1 Behavior aong the horizonta joints... 1 2.1.2 Structura design parameters... 11 ii
2.1.3 Base-shear-roo-drit reationship... 11 2.1.3.1 Decompression state... 11 2.1.3.2 Sotening state... 12 2.1.3.3 Yieding state... 13 2.1.3.4 Faiure state... 13 2.2 Prestressed Beams with Openings... 13 2.3 Monoithic Cast-in-Pace Concrete Was with Openings... 13 2.4 Precast Concrete Was with Openings... 14 2.5 Fiber Eement Mode... 14 2.6 Cosed-Form Soutions or Ininite Eastic Panes with Openings... 15 CHAPTER 3 PARAMETRIC INVESTIGATION... 19 3.1 Opening Dimensions, o and h o... 19 3.2 Pane Dimensions, p, h p, and t p... 2 3.3 Initia Concrete Stress, ci... 22 3.4 Post-Tensioning Loads... 23 3.5 Gravity Loads... 23 3.6 Spira Coninement... 23 3.7 Anaysis o the Was... 24 CHAPTER 4 ANALYTICAL MODEL... 25 4.1 Modeing o the Post-Tensioning Bars... 25 4.2 Modeing o the Wa Panes... 27 iii
4.2.1 Reinement o the inite eement mesh... 29 4.3 Modeing o Concrete in Compression... 29 4.4 Concrete Faiure Enveope under Biaxia Loading... 3 4.4.1 ABAQUS biaxia aiure ratios... 32 4.4.2 ABAQUS aiure ratio 1... 33 4.4.3 ABAQUS aiure ratio 2... 33 4.4.4 ABAQUS aiure ratio 3... 34 4.4.5 ABAQUS aiure ratio 4... 35 4.4.6 Faiure ratios used to mode the was... 36 4.5 Modeing o Bonded Mid Stee Reinorcement... 37 4.6 Modeing o Behavior Aong the Horizonta Joints... 37 4.7 Modeing o Gravity Loads... 39 4.8 Modeing o Latera Loads... 39 4.9 Veriication o the Anaytica Mode... 39 4.9.1 Comparisons with the iber eement mode... 39 4.9.2 Comparisons with cosed-orm soutions... 4 CHAPTER 5. DESIGN OF PANEL REINFORCEMENT UNDER VERTICAL LOADS ONLY... 42 5.1 Critica Pane Regions under Vertica Loads... 42 5.2 Design o Pane Reinorcement: Overview... 45 5.3 Truss Mode... 45 5.3.1 Estimation o Pane Top Stresses... 47 iv
5.3.2 Estimation o C r and xg p... 51 5.3.3 Estimation o Side Chord Stresses and xg s... 51 5.4 Pacement o the Pane Reinorcement... 52 5.5 Design o Upper Story Panes... 55 5.6 Base Pane Reinorcement Resuts... 59 5.6.1 Resut averages... 63 CHAPTER 6 DESIGN OF PANEL REINFORCEMENT UNDER COMBINED VERTICAL AND LATERAL LOADS... 64 6.1 Eect o Openings on the Latera Load Behavior o the Was... 64 6.2 Critica Pane Regions... 65 6.3 Determining the Critica Pane Sections or Design: Overview... 69 6.4 Was Without Openings... 69 6.4.1 Contact ength at the bottom o the base pane... 7 6.4.2 Contact ength at the top o the base pane... 71 6.5 Was With Openings... 72 6.5.1 Contact ength at the bottom o the base pane... 72 6.5.2 Large openings versus sma openings... 74 6.5.3 Contact ength at the top o the base pane... 76 6.6 Critica Section at the Bottom o the Base Pane... 77 6.7 Critica Section at the Top o the Base Pane... 77 6.8 Overview o the Proposed Design Approach... 78 6.9 Design o Pane Top Reinorcement... 8 v
6.9.1 Forces at the top o the pane... 81 6.9.1.1 Axia orce, N pt... 82 6.9.1.2 Shear orce, V pt... 85 6.9.2 Forces in the tension side chord... 88 6.9.2.1 Axia orce, N 2... 89 6.9.2.2 Shear orce, V 2... 92 6.9.2.3 Moment, M 2m... 95 6.9.3 Design tension orce and required reinorcement... 99 6.1 Design o Pane Side Reinorcement... 1 6.11 Design o Pane Bottom Reinorcement... 1 6.11.1 Forces in the compression side chord... 1 6.11.2 Forces in Section 4... 11 6.11.3 Design tension orce and required reinorcement... 12 6.12 Design o Reinorcement Around Openings... 14 6.13 Pacement o the Pane Reinorcement... 14 6.14 Design o Shear Reinorcement... 15 6.15 Roo-Drit at the Faiure State... 16 6.16 Design o the Upper Story Panes... 16 6.16.1 Required reinorcement in the upper story panes... 18 6.17 Base Pane Reinorcement Resuts... 112 6.17.1 Pane top reinorcement, D c1b... 113 6.17.2 Pane side reinorcement, D c2b... 117 vi
6.17.3 Pane bottom reinorcement, D c3b... 121 6.17.4 Resut averages... 125 CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS... 126 REFERENCES... 129 APPENDIX A DESIGN EXAMPLE, VERTICAL LOADS ONLY... 132 APPENDIX B DESIGN EXAMPLE, COMBINED VERTICAL AND LATERAL LOADS... 136 vii
LIST OF TABLES TABLE 2.1 VALUES OF k... 17 TABLE 3.1. PARAMETRIC INVESTIGATION... 2 TABLE 3.2 POST-TENSIONING AND GRAVITY LOADS... 22 TABLE 3.3 CONCRETE PARAMETERS... 24 TABLE 5.1 REQUIRED REINFORCEMENT AT TOP AND BOTTOM OF OPENING... 59 TABLE 5.2 AVERAGE OF PREDICTED/ REINFORCEMENT RATIO... 61 TABLE 6.1 CONTACT LENGTH AT THE BOTTOM OF THE BASE PANEL... 71 TABLE 6.2 CONTACT LENGTH AT THE TOP OF THE BASE PANEL... 73 TABLE 6.3 CRITICAL SECTION AT THE BOTTOM OF THE BASE PANEL... 78 TABLE 6.4 CRITICAL SECTION AT THE TOP OF THE BASE PANEL... 79 TABLE 6.5 REQUIRED REINFORCEMENT AT THE TOP OF THE BASE PANEL... 113 TABLE 6.6 REQUIRED REINFORCEMENT AT THE SIDE OF THE BASE PANEL... 117 TABLE 6.7 REQUIRED REINFORCEMENT AT THE BOTTOM OF THE BASE PANEL... 121 viii
TABLE 6.8 AVERAGE OF PREDICTED / REINFORCEMENT RATIO... 125 ix
LIST OF FIGURES Figure 1.1 Unbonded post-tensioned precast concrete wa: (a) eevation; (b) cross-section near base... 2 Figure 1.2 Wa under atera oads: (a) dispaced shape; (b) stresses on the base pane... 3 Figure 1.3 Wa with openings... 4 Figure 1.4 Cracking in a wa pane... 5 Figure 2.1 Wa WH1M: (a) eevation; (b) cross-section near base (ha the wa ength)... 9 Figure 2.2 Prototype buiding pan ayout... 1 Figure 2.3 Base-shear versus roo-drit reationship o Wa WH1M... 12 Figure 2.4 Ininite eastic pane with an opening... 16 Figure 3.1 Eevation and cross-section o the parametric was: (a) p =2 t, ( =.29; (b) p =2 t, ( =.18; (c) p =2 t, ( =.11; (d) p =2 t, ( =.57; (e) p =15 t, ( =.18; () p =12 t, ( =.18... 21 Figure 4.1 Finite eement mesh: (a) reined mesh with openings; (b) coarse mesh... 26 Figure 4.2 Stress-strain reationship o the post-tensioning stee... 27 Figure 4.3 Concrete stress-strain reationships... 3 Figure 4.4 Concrete biaxia aiure enveope... 31 Figure 4.5 The 9x9 inite eement mode... 32 Figure 4.6 Faiure ratio 1: (a) biaxia aiure enveope (b) stress-strain reationships... 33 x
Figure 4.7 Eect o aiure ratio 2 on the concrete biaxia aiure enveope... 34 Figure 4.8 Faiure ratio 3: (a) biaxia aiure enveope; (b) stress-strain reationships... 35 Figure 4.9 Eect o aiure ratio 4 on the concrete biaxia aiure enveope... 35 Figure 4.1 ABAQUS versus Mander concrete stress-strain reationships... 36 Figure 4.11 Deormed inite eement mode: (a) entire wa; (b) base pane... 38 Figure 4.12 Veriication o the inite eement mode: (a) base-shear-roo-drit reationship; (b) contact ength... 4 Figure 4.13 Finite eement mode versus cosed-orm soutions... 41 Figure 5.1 Stress contours in the base pane: (a) maximum principe stresses (b) minimum principe stresses... 43 Figure 5.2 Stress contours in the base pane: (a) horizonta stresses; (b) vertica stresses... 44 Figure 5.3 Pane stresses... 46 Figure 5.4 Design o pane reinorcement: (a) truss mode; (b) truss mode enarged... 47 Figure 5.5 Estimation o p : (a) bottom two panes; (b) p versus 2 c... 48 Figure 5.6 Pane top stresses: (a) ( h =.25, ( =.29, p =2 t; (b) ( =.2, ( =.29, p =2 t; (c) ( =.2, ( h =.25, ( =.18; (d) ( =.2, ( h =.25, p =2 t... 5 Figure 5.7 Side chord stresses: (a) ( h =.25, ( =.29, p =2 t; (b) ( =.2, ( =.29, p =2 t;(c) ( =.2, ( h =.25, ( =.18; (d) ( =.2, ( h =.25, p =2 t..... 53 Figure 5.8 Pane reinorcement exampe... 54 Figure 5.9 Principe stress contours in Wa WH1M (( =.29, p =2 t): (a) without opening; (b) with opening (( =.4, ( h =.38)... 55 Figure 5.1 Horizonta stress contours in Wa WH1M (( =.29, p =2 t): (a) without opening; (b) with opening (( =.4, ( h =.38)... 57 xi
Figure 5.11 Vertica stress contours in Wa WH1M (( =.29, p =2 t): (a) without opening; (b) with opening (( =.4, ( h =.38)... 58 Figure 5.12 Eect o ( on D v : (a) ( h =.25, p =2 t; (b) ( h =.25, ( =.18; (c) ( =.57, p =2 t; (d) ( =.29, p =2 t... 6 Figure 5.13 Eect o ( h on D v : (a) ( =.3, p =2 t; (b) ( =.3, ( =.18; (c) ( =.57, p =2 t; (d) ( =.29, p =2 t... 61 Figure 5.14 Eect o p on D v : (a) ( =.3, ( =.18; (b) ( h =.25, ( =.18... 62 Figure 5.15 Eect o ( on D v : (a) ( =.3, p =2 t; (b) ( h =.25, p =2t... 62 Figure 6.1 Base-shear-roo-drit reationship o was without and with openings... 65 Figure 6.2 Stress contours under atera oads: (a) maximum principe stresses (b) minimum principe stresses... 66 Figure 6.3 Stress contours in the base pane: (a) horizonta stresses; vertica stresses... 68 Figure 6.4 Pane reinorcement... 69 Figure 6.5 Estimation o c b... 7 Figure 6.6 Estimation o c t N... 72 Figure 6.7 Contact ength at the base o was without and with openings... 74 Figure 6.8 Opening size actor: (a) sma opening; (b) arge opening... 75 Figure 6.9 Contact ength at top o the base pane o a wa with and without openings... 76 Figure 6.1 Design orces: (a) critica sections; (b) ree body diagram or Sections 1 and 2; (c) ree body diagram or Sections 2 and 3... 8 Figure 6.11 Eect o ( on N pt : (a) ( h =.25, p =2 t; (b) ( h =.25, ( =.18; (c) ( =.29, p =2 t... 82 Figure 6.12 Eect o ( h on N pt : (a) ( =.3, p =2 t; (b) ( =.3, ( =.18; (c) ( =.29, p =2 t... 83 Figure 6.13 Eect o p on N pt : (a) ( =.3, ( =.18, (b) ( h =.25, ( =.18... 84 xii
Figure 6.14 Eect o ( on N pt : (a) ( =.3, p =2 t; (b) ( h =.25, p =2 t... 84 Figure 6.15 Eect o ( on V pt : (a) ( h =.25, p =2 t; (b) ( h =.25, ( =.18; (c) ( =.29, p =2 t... 85 Figure 6.16 Eect o ( h on V pt : (a) ( =.3, p =2 t; (b) ( =.3, ( =.18; (c) ( =.29, p =2 t... 86 Figure 6.17 Eect o p on V pt : (a) ( =.3, ( =.18, (b) ( h =.25, ( =.18... 87 Figure 6.18 Eect o ( on V pt : (a) ( =.3, p =2 t; (b) ( h =.25, p =2 t... 87 Figure 6.19 Eect o ( on N 2 : (a) ( h =.25, p =2 t; (b) ( h =.25, ( =.18; (c) ( =.29, p =2 t... 89 Figure 6.2 Eect o ( h on N 2 : (a) ( =.3, p =2 t; (b) ( =.3, ( =.18; (c) ( =.29, p =2 t... 9 Figure 6.21 Eect o p on N 2 : (a) ( =.3, ( =.18, (b) ( h =.25, ( =.18... 91 Figure 6.22 Eect o ( on N 2 : (a) ( =.3, p =2 t; (b) ( h =.25, p =2 t... 91 Figure 6.23 Eect o ( on V 2 : (a) ( h =.25, p =2 t; (b) ( h =.25, ( =.18; (c) ( =.29, p =2 t... 92 Figure 6.24 Eect o ( h on V 2 : (a) ( =.3, p =2 t; (b) ( =.3, ( =.18; (c) ( =.29, p =2 t... 93 Figure 6.25 Eect o p on V 2 : (a) ( =.3, ( =.18, (b) ( h =.25, ( =.18... 94 Figure 6.26 Eect o ( on V 2 : (a) ( =.3, p =2 t; (b) ( h =.25, p =2 t... 94 Figure 6.27 Frame structure... 95 Figure 6.28 Eect o ( on M 2m : (a) ( h =.25, p =2 t; (b) ( h =.25, ( =.18; (c) ( =.29, p =2 t... 96 Figure 6.29 Eect o ( h on M 2m : (a) ( =.3, p =2 t; (b) ( =.3, ( =.18; (c) ( =.29, p =2 t... 97 Figure 6.3 Eect o p on M 2m : (a) ( =.3, ( =.18, (b) ( h =.25, ( =.18... 98 Figure 6.31 Eect o ( on M 2m : (a) ( =.3, p =2 t; (b) ( h =.25, p =2 t... 98 xiii
Figure 6.32 Estimation o T... 99 Figure 6.33 Estimation o T 2... 1 Figure 6.34 Axia stress distribution in the compression chord... 11 Figure 6.35 Shear stress distribution in the compression chord... 12 Figure 6.36 Estimation o T 3... 13 Figure 6.37 Principe stress contours in Wa WH1M (( =.29, p =2 t); (a) without openings; (b) with openings (( =.4, ( h =.38)... 17 Figure 6.38 Horizonta stress contours in Wa WH1M (( =.29, p =2 t); (a) without openings; (b) with openings (( =.4, ( h =.38)... 19 Figure 6.39 Vertica stress contours in Wa WH1M (( =.29, p =2 t); (a) without openings; (b) with openings (( =.4, ( h =.38)... 11 Figure 6.4 Required reinorcement in the upper story panes... 112 Figure 6.41 Eect o ( on D c1b : (a) ( h =.25, p =2 t; (b) ( h =.25,( =.18; (c) ( =.29, p =2 t... 114 Figure 6.42 Eect o ( h on D c1b : (a) ( =.3, p =2 t; (b) ( =.3,( =.18; (c) ( =.29, p =2 t... 115 Figure 6.43 Eect o p on D c1b : (a) ( =.3, ( =.18; (b) ( h =.25, ( =.18... 116 Figure 6.44 Eect o ( on D c1b : (a) ( =.3, p =2 t; (b) ( h =.25, p =2 t... 116 Figure 6.45 Eect o ( on D c2b : (a) ( h =.25, p =2 t; (b) ( h =.25,( =.18; (c) ( =.29, p =2 t... 118 Figure 6.46 Eect o ( h on D c2b : (a) ( =.3, p =2 t; (b) ( =.3,( =.18; (c) ( =.29, p =2 t... 119 Figure 6.47 Eect o p on D c2b : (a) ( =.3, ( =.18; (b) ( h =.25, ( =.18... 12 Figure 6.48 Eect o ( on D c2b : (a) ( =.3, p =2 t; (b) ( h =.25, p =2 t... 12 Figure 6.49 Eect o ( on D c3b : (a) ( h =.25, p =2 t; (b) ( h =.25,( =.18; (c) ( =.29, p =2 t... 122 xiv
Figure 6.5 Eect o ( h on D c3b : (a) ( =.3, p =2 t; (b) ( =.3,( =.18; (c) ( =.29, p =2 t... 123 Figure 6.51 Eect o p on D c3b : (a) ( =.3, ( =.18; (b) ( h =.25, ( =.18... 124 Figure 6.52 Eect o ( on D c3b : (a) ( =.3, p =2 t; (b) ( h =.25, p =2 t... 124 xv
LIST OF SYMBOLS a = variabe in cosed-orm soution ag = variabe in cosed-orm soution a b N = ength o rectanguar stress bock at bottom o base pane without opening A c1b = area o reinorcing stee at top o base pane, combined oading A c2b = area o reinorcing stee at sides o base pane, combined oading A c3b = area o reinorcing stee at bottom o base pane, combined oading A c4b = area o reinorcing stee at top and bottom o opening in base pane, combined oading A c5b = area o reinorcing stee at sides o opening in base pane, combined oading A g = gross cross-section area o wa A min = area o minimum reinorcement (2 #5 bars) a p = area o an individua post-tensioning bar a pj = irst coeicient or pane top stress distribution in region j A v = area o reinorcing stee at top and bottom o opening, vertica oads ony b pj = second coeicient or pane top stress distribution in region j C 1 = compression orce in Section 1 C 2 = compression orce in Section 2 C 3 = compression orce in Section 3 xvi
c b = contact ength at bottom o base pane with opening c b N = contact ength at bottom o base pane without opening c be = distance rom end o wa to edge o contact region at base o wa c pj = third coeicient or pane top stress distribution in region j C r = compression orce in truss mode c t = contact ength at top o base pane with opening c t N = contact ength at top o base pane without opening C v = compression orce at top o opening under vertica oads ony d c = eective depth o horizonta chord measured rom compression edge to ongitudina reinorcement in tension 11 = norma stress aong 1-1 axis 22 = norma stress aong 2-2 axis c = stress in concrete c N = compressive strength o unconined concrete cc = compressive strength o spira conined concrete ci = initia stress in concrete due to gravity and post-tensioning p = compression stress appied to top o pane p = compression stress at top o pane at centerine p1 = compression stress at top o pane at x 1 p2 = compression stress at top o pane at x 2 pa = uniormy distributed stress transerred to pane rom pane above pe = compression stress at top o pane at edge based on a inear stress distribution xvii
pi = initia stress in post-tensioning bar pp = uniormy distributed stress due to gravity oads appied on oor eve above pane pu = utimate strength o post-tensioning stee py = yied strength o post-tensioning stee se = compression stress in side chord at edge o pane sr = compression stress in side chord at x r tm = maximum tension stress at top o opening y = yied strength o mid stee G a = sum o gravity oads appied on upper story panes G b = tota gravity oad acting at base o wa G p = gravity oad appied on oor eve above pane G pt = resutant o uniormy distributed gravity oad appied et o critica section at pane top h c = height o horizonta chord in base pane h cu = height o horizonta chord in upper story panes h o = height o opening h p = height o base pane h pu = height o upper story pane h t1 = height o tension zone in Section 1 h t3 = height o tension zone in Section 3 h tv = height o tension zone at top o opening, vertica oads ony k = characteristic o opening aspect ratio c = ength o side chord xviii
o = ength o opening p = ength o pane t2 = ength o tension zone in Section 2 M 2 = moment o axia orce in Section 2 about midde o Section 2 M 2m = moment o axia orce in Section 2 about pane centerine M 3 = moment o axia orce in Section 3 about center o Section 3 M 4 = moment o axia orce in Section 4 about center o Section 4 M p2 = moment acting in base pane at bottom o opening M pb = moment acting at base o wa due to atera oads and post-tensioning M pt = moment acting at top o base pane due to atera oads and post-tensioning M r = moment o axia orce in compression side chord about midde o chord n = story number N 2 = axia orce in Section 2 N 3 = axia orce in Section 3 N 4 = axia orce in Section 4 n max = tota number o stories N pb = tota axia orce at base o wa due to gravity and post-tensioning N pt = axia orce transerred to top o base pane rom pane above N pt = axia orce transerred to base pane at et o critica section at pane top N r = axia orce in compression side chord P = sum o orces in post-tensioning bars at aiure P i = tota initia post-tensioning orce xix
t c = thickness o spira conined concrete t p = thickness o pane s = opening size actor T 1 = tension orce in Section 1 T 2 = tension orce in Section 2 T 3 = tension orce in Section 3 T v = tension orce occurring at top o opening under vertica oads ony U = constant in cosed-orm soution V = shear orce appied at top o rame V 2 = shear orce in Section 2 V 4 = shear orce in Section 4 V p = shear orce appied at top o base pane V pb = shear orce at base o wa V pt = shear orce transerred to top o base pane rom pane above V pt = shear orce transerred to base pane at et o critica section at pane top V r = shear orce in compression side chord x = horizonta distance measured rom pane centerine x 1 = ocation where F p1 changes to F p2 x 2 = ocation where F p2 changes to F p3 x cb = distance o Section 3 rom compression end o pane x ct = distance o Section 1 rom compression end o pane xg p = resutant ocation o C r at top o pane xx
x r = distance over which stresses are summed to determine C r x s = distance used to deine a arge opening xg s = resutant ocation o C r at top o side chord y c3 = ocation o compression stress resutant in Section 3 " = actor or magnitude o equivaent rectanguar stress bock $ 1 = actor or depth o equivaent rectanguar stress bock ( = ratio o ci to c N ( h = ratio o h o to h p ( = ratio o o to p, c = strain in concrete, cc = strain in spira conined concrete when maximum compressive stress is reached, cu = utimate strain capacity o spira conined concrete 2 = ange (in radians) around edge o opening or cosed-orm soution 2 c = ange to determine p (in degrees) 2 t = ange o truss mode (in degrees) > a = height to ength ratio o base pane > h = horizonta chord height actor > = vertica chord ength actor D c1b = reinorcement ratio at top o base pane, combined oading D c1n = reinorcement ratio at top o n th pane, combined oading D c2b = reinorcement ratio at sides o base pane, combined oading D c2n = reinorcement ratio at sides o n th pane, combined oading xxi
D c3b = reinorcement ratio at bottom o base pane, combined oading D c3n = reinorcement ratio at bottom o n th pane, combined oading D c4b = reinorcement ratio at top and bottom o opening in base pane, combined oading D c4n = reinorcement ratio at top and bottom o opening in n th pane, combined oading D c5b = reinorcement ratio at sides o opening in base pane, combined oading D c5n =reinorcement ratio at sides o opening in n th pane, combined oading D min,h = minimum reinorcement ratio in horizonta chords D min,v = minimum reinorcement ratio in vertica chords D sp = spira reinorcement ratio D v = reinorcement ratio at top and bottom o opening, vertica oads ony F p = stress distribution at top o pane, vertica oads F p1 = second order pane top stress distribution in region 1 F p2 = second order pane top stress distribution in region 2 F p3 = second order pane top stress distribution in region 3 F pj = second order pane top stress distribution in region j F s = stress distribution in side chord, vertica oads F s1 = second order side chord stress distribution in region 1 F s2 = irst order side chord stress distribution in region 2 F 2 = stress around edge o opening H = unction o stress appied to ininite pane and ocation around opening edge S = stress distribution actor T = stresses unction around edge o opening xxii
n = stress unction around edge o opening n o = stress unction or boundary conditions around edge o opening xxiii
ACKNOWLEDGMENTS The research was unded by a PCI Danie P. Jenny Research Feowship and by the University o Notre Dame. The support o the Precast/Prestressed Concrete Institute and the University o Notre Dame is grateuy acknowedged. The support o the PCI Research Director P. Joha and Past Chairman o the Research and Deveopment Committee H. Widen, H. Widen & Associates, Inc. is grateuy acknowedged. The support and advice provided by the PCI Ad Hoc Advisory Group on the Feowship is acknowedged: N. Ceand, Bue Ridge Design, Inc.; T. D'Arcy, The Consuting Engineers Group, Inc.; J. Homan, Sturm Engineering Company; and S. Pessiki, Lehigh University. The researchers aso wish to thank two other individuas or their contributions to the project: K. Baur, High Concrete Structures, Inc. and R. Sause, Lehigh University. M. Nieted heped in the deveopment o the inite eement mode during the summer o 1998 through the Research Experiences or Undergraduates (REU) Program unded by the Nationa Science Foundation. The support o the Nationa Science Foundation or the REU Program is acknowedged. The opinions, indings, and concusions expressed in this report are those o the authors and do not necessariy reect the views o the individuas and organizations acknowedged above. xxiv
CHAPTER 1 INTRODUCTION 1.1 Overview Precast concrete is a widey used type o construction throughout the word. Its cost eectiveness and ast production enabe innovation in both design and construction. Due to the high technoogy and ow abor cost invoved with precast concrete structures, this method matches the strengths and conditions o the construction industry in the United States. The deveopment o precast concrete structura systems in the United States has not been as extensive as in some other countries (Priestey 1991). One o the major actors inhibiting the deveopment o precast construction in the U.S. is the uncertainty in the behavior o precast concrete structures in seismic regions. A ack o experience on the seismic response o muti-story structura systems to strong earthquakes and itte experimenta data have resuted in precast systems being ess common in seismic regions in the U.S. Current U.S. mode buiding codes (e.g., NEHRP-97, IBC 2) aso restrict the innovation o precast concrete systems in seismic regions. In the absence o experimenta data, the codes require that precast concrete systems emuate the behavior o monoithic cast-in-pace reinorced concrete systems. Extensive testing is required or the design o non-emuative precast systems. This requirement or testing, because it is expensive and time consuming, inhibits the deveopment o new and innovative systems. This has caused most previous research on precast concrete to ocus on the emuation o monoithic cast-in-pace concrete. However, in many cases, precast concrete systems that emuate monoithic concrete systems perorm poory in seismic regions because o probems in the joint regions between the precast members (Kurama et a. 1996). Furthermore, emuation reduces the economic advantages o precast concrete and imits the reaization o the true potentia o these systems. In order to deveop recommendations or the use o precast concrete systems in seismic regions, the PREcast Seismic Structura Systems (PRESSS) Research Program was initiated (Priestey 1991) with unding rom the Nationa Science Foundation (NSF), the Precast/Prestressed Concrete Institute (PCI), and the Precast/Prestressed Concrete Manuacturers Association o Caiornia, Inc. (PCMAC). 1
The main objectives o the PRESSS Program were to deveop recommendations or the seismic design o precast concrete structura systems and to deveop new systems. The research program was conducted in three phases. Phase I ocused on identiying and evauating the most promising seismic systems. Phase II invoved anaytica and experimenta studies o the systems seected in Phase I. Finay, Phase III investigated the design, erection, testing, and anaysis o a ive-story precast concrete buiding (Priestey et a. 1999). 1.2 Unbonded Post-Tensioned Precast Concrete Was Previous research conducted as a part o Phase II and Phase III o the PRESSS Program has shown that unbonded post-tensioned precast concrete was may oer signiicant advantages as primary atera oad resisting systems in seismic regions (Kurama et a. 1999a, 1999b, Priestey et a. 1999). These was do not emuate the behavior o monoithic cast-in-pace reinorced concrete was. As an exampe, Figure 1.1 shows the eevation and cross-section near the base o a six-story unbonded post-tensioned precast concrete wa. The wa is constructed by joining precast wa panes across horizonta joints using post-tensioning bars which are anchored to the wa ony at the roo and at the oundation. atera oad gravity oad PT anchorage PT bar (unbonded) wa pane horizonta joint spira reinorcement oundation PT bar (a) wire mesh spira reinorcement (b) Figure 1.1 Unbonded post-tensioned precast wa: (a) eevation; (b) cross-section near base 2
The post-tensioning bars are paced inside oversized ducts to aow the wa to dispace ateray without resuting in the kinking o the bars. No grout is paced inside the ducts, and thus the posttensioning bars are not bonded to the wa panes. Dry-pack or grout may be used between the panes or construction toerances and or aignment purposes. Spira reinorcing stee is used to conine the concrete near the base o the wa. Wire mesh is used as bonded reinorcement in the panes. Unbonded post-tensioned precast was resist gravity oads as we as atera oads. During a seismic event, the precompression orces rom post-tensioning and gravity may be overcome, eading to the ormation o gaps aong the horizonta joints between the panes and between the wa and the oundation. Figure 1.2(a) shows an exampe o this behavior or a wa subjected to vertica (due to post-tensioning and gravity) and atera oads. The presence o the gaps aong the horizonta joints causes the compression stresses due to post-tensioning and gravity and the shear stresses due to atera oads to be distributed over the ength o each pane that is in contact with the oundation or with the adjacent panes. Figure 1.2(b) gives an exampe o these contact stresses at the top and bottom o the base pane. compression stresses due to PT and gravity (a) (b) shear stresses due to atera oads discrete gap Figure 1.2 Wa under atera oads: (a) dispaced shape; (b) stresses on the base pane 1.3 Was with Openings The previous research on the seismic behavior and design o unbonded post-tensioned was is imited to was without openings. However, the use o openings in the wa panes as shown in Figure 1.3 may be desirabe. For exampe, openings may be needed to accommodate windows, 3
doors, and mechanica penetrations due to unctiona and/or architectura requirements. Large openings may aso be necessary in precast concrete parking structures or security reasons. atera oad gravity oad opening Figure 1.3 Wa with openings The seismic design o unbonded post-tensioned precast concrete was can be done using a procedure which was deveoped previousy or was without openings. This design procedure, which is described in detai by Kurama et a. (1999a, 1999b), can be used to determine the wa ength and thickness as we as the required amount o post-tensioning and spira reinorcement. The use o openings in the wa panes is not addressed by the previous design approach. These openings can resut in arge tension stresses, and thus, cracking in the panes under vertica and atera oads. The gaps which orm aong the horizonta joints under atera oads may aso cause cracking in the panes. These cracks can imit the vertica and atera oad carrying capacity o the was by causing premature aiure o the wa panes. Thus, bonded mid stee reinorcement may be needed in the panes to imit the size o the cracks. Critica ocations in a precast wa pane where cracks may orm under vertica (due to posttensioning and gravity) and atera oads are shown in Figure 1.4. The atera oads are assumed to be acting rom et to right. The opening can cause cracking in the pane under the presence o vertica oads ony. This cracking occurs at the center o the top and bottom edges o the opening. The design o the pane reinorcement required to imit the size o these cracks is discussed in Chapter 5. 4
Figure 1.4 Cracking in a wa pane During a seismic event, cracking may occur at the sides o the pane and near the corners o the opening. Cracking may aso occur at the top and bottom o the pane due to the ormation o gaps aong the horizonta joints. The determination o the critica pane regions where cracking may occur under combined vertica and atera oads and the design o the reinorcement required to imit the size o these cracks are discussed in Chapter 6. 1.4 Objectives The objectives o the research described in this report are: 1) To determine the eect o the openings on the behavior o the was and the wa panes; 2) To determine the critica ocations in the wa panes and the required amount o pane reinorcement; 3) To deveop design methods or the pane reinorcement. 1.5 Scope and Approach The research is conducted as a part o a Danie P. Jenny Research Feowship unded by the Precast/Prestressed Concrete Institute. The project ocuses on two stages o oading or the was. In the irst stage, the was are subjected to vertica orces ony, such as due to gravity oads and posttensioning. This is the oading stage which a typica wa woud be subjected to during most o its service ie. The second stage o oading is the combination o vertica oads with atera oads, such as due to earthquakes. The behavior and design o the was in this stage are investigated based on a series o noninear static push-over anayses. The inite eement program, ABAQUS (Hibbitt et a. 1998) is used to mode the was. 5
For each oading stage, the critica regions in the wa panes are identiied and a design approach to determine the required amount o bonded pane reinorcement to contro the cracks and prevent premature aiure o the was is proposed. The eect o opening ength and height, wa ength, and initia stress in the concrete due to post-tensioning and gravity oads on the behavior and design o the was is investigated. The number o stories is kept constant at six stories because the design o the pane reinorcement is not expected to be inuenced signiicanty by the number o stories. It is assumed that horizonta joints exist at each oor eve and at the oundation since arger pane sizes may cause diicuties in the transportation o the panes to the construction site due to increased weight and/or size, particuary or onger was. Ony rectanguar openings which are ocated at the center o the wa panes are considered. It is assumed that each wa pane contains an opening and that the opening size in a wa is constant. To achieve the research objectives, the oowing approach was taken: 1) A inite eement mode was deveoped or the was using the ABAQUS Program (Hibbitt et a. 1998). The mode was veriied by comparing the resuts with a second anaytica mode deveoped during previous research and with cosed-orm anaytica soutions or eastic panes with openings. 2) A series o parametric was were determined. The parameters which were varied are: (1) ength o the openings; (2) height o the openings; (3) ength o the panes; and (4) initia stress in the concrete due to post-tensioning. 3) Finite eement anayses were perormed on the parametric was to investigate the behavior and design o the panes under vertica oads (due to gravity and post-tensioning) ony. 4) A design approach was deveoped to determine the required bonded mid stee reinorcement in the panes under vertica oads. 5) Finite eement anayses were perormed on the parametric was to investigate the behavior and design o the panes under combined vertica and atera oads. 6) A design approach was deveoped to determine the required bonded mid stee reinorcement in the panes under combined vertica and atera oads. 6
1.6 Organization o Report The remainder o this report is organized as oows: Chapter 2 discusses the previous research on unbonded post-tensioned precast concrete was, concrete members with openings, and cosed-orm anaytica soutions or stresses around rectanguar openings in ininite eastic panes. Chapter 3 taks about the parameters studied by the research and the parametric was. Chapter 4 discusses the deveopment and veriication o the inite eement mode used to investigate the behavior and design o the was. Chapter 5 discusses the behavior and design o the was under vertica oads ony and proposes a method or the design o the required reinorcement in the panes. Chapter 6 discusses the behavior and design o the was under combined vertica and atera oads and proposes a method or the design o the required reinorcement in the panes. Chapter 7 presents a summary and concusions o the research. 7
CHAPTER 2 BACKGROUND This chapter provides background inormation or the report. An overview o the previous research on the seismic behavior and design o unbonded post-tensioned precast concrete was is presented. Then, previous research on concrete members and structures, both cast-in-pace and precast, with openings is discussed. Finay, avaiabe cosed-orm anaytica soutions which can be used to determine the stresses around rectanguar openings in ininite eastic panes are summarized. 2.1 Unbonded Post-Tensioned Precast Concrete Was Previous research by Kurama et a. (1996, 1999a, 1999b) and Priestey et a. (1999) have shown the advantages o unbonded post-tensioned precast concrete was or seismic regions. As an exampe, Figure 2.1 shows the eevation and cross-section near base o a six-story prototype wa reerred to as Wa WH1M which was designed or a six-story oice buiding (with a pan ayout as shown in Figure 2.2) in a region with high seismicity (e.g., Western U.S.) and a site with a medium soi proie (Kurama et a. 1999a). The wa consists o six precast concrete panes with horizonta joints at the oor eves and at the oundation. The wa panes are connected using post-tensioning bars which are paced inside oversized ducts as shown in Figure 2.1. The ducts aow the post-tensioning bars to be paced in the wa ater the panes are erected. The post-tensioning bars are prestressed to pi =.6 pu where pu =16 ksi is the utimate strength o the post-tensioning stee. The ducts are not grouted ater post-tensioning in order to prevent bond rom orming between the bars and the concrete. The panes do not have any bonded reinorcement connecting them aong the horizonta joints. During a seismic event, the ack o bonded reinorcement across the horizonta joints aows the panes to orm gaps as shown in Figure 1.2. The unbonded post-tensioning bars resist the atera oads and contro the size o the gaps during the earthquake. Ater the earthquake, the posttensioning bars provide a restoring orce that pus the wa back towards its origina undeormed position. 8
atera oads gravity oads 81 t spira reinorcement 2 t (a) C L a =1.49 in. 2 p 1 in. 94 88 82 76 7 64 12 in. #3 Spiras ρ = 7.3% sp pi =.6pu 12 in. (b) Figure 2.1 Wa WH1M: (a) eevation; (b) cross-section near base (ha the wa ength) 9
8 x 24 t. = 192 t. 4 t. 3 t. 4 t. hoowcore panes coumn gravity oad rame L-beam atera oad rame wa inverted T-beam N S Figure 2.2 Prototype buiding pan ayout Unbonding o the post-tensioning stee has two important advantages: (1) it resuts in a uniorm strain distribution in the stee, and thus, deays the noninear straining (i.e., yieding) o the posttensioning bars; (2) it signiicanty reduces the amount o tensie stresses in the concrete, thus reducing damage due to cracking. The strength o the unconined concrete, c 1 in Wa WH1M is assumed to be 6 ksi. During a seismic event, arge compression stresses orm in the bottom corners o the base (i.e., irst story) pane due to the ormation o a gap aong the base pane to oundation joint. To increase the compressive strength and ductiity o the concrete in these regions, spira reinorcement is used to conine the concrete aong the height o the base pane as shown in Figure 2.1. The spira reinorcement ratio (i.e., voume o spira reinorcing stee divided by voume o conined concrete core) near the base o Wa WH1M is! sp =7.3%. The eect o the spira reinorcement on the concrete is discussed in Chapter 4. 2.1.1 Behavior aong the horizonta joints During a seismic event, two modes o dispacement can occur aong the horizonta joints o an unbonded post-tensioned precast wa. The irst mode is the ormation o gaps as shown in Figure 1.2. The second mode is shear sip. Previous research (Kurama et a. 1996, 1999a, 1999b) has shown that shear sip aong the horizonta joints is not desired. Thus, the was are designed not to have sip aong the joints (Kurama et a. 1996, 1999a, 1999b). 1
2.1.2 Structura design parameters The seismic response characteristics o unbonded post-tensioned precast concrete was depend on their design. Kurama et a. (1996) identiied and studied the eect o eeven structura design parameters on the atera oad behavior o the was. A design approach or the was was deveoped using these parameters which are: (1) eccentricity o the post-tensioning bars; (2) unbonded ength o the post-tensioning bars; (3) initia stress in the post-tensioning bars; (4) area o the posttensioning bars; (5) initia stress in the concrete due to post-tensioning; (6) initia stress in the concrete due to gravity oads; (7) strength o unconined concrete; (8) amount o spira reinorcement; (9) ength o the wa; (1) thickness o the wa; and (11) aspect ratio o the wa cross-section (i.e. wa ength divided by wa thickness). 2.1.3 Base-shear-roo-drit reationship During a seismic event, an unbonded post-tensioned precast wa experiences atera dispacements as shown in Figure 1.2. Figure 2.3 shows the noninear base-shear-roo-drit reationship o Wa WH1M under combined atera and vertica oads. The base-shear is equa to the sum o the atera oads appied to the wa. The roo-drit is determined by dividing the horizonta dispacement o the roo with the wa height. The atera oads are appied at the oor and roo eves and are distributed in a trianguar pattern over the height o the wa as shown in Figure 2.1. Kurama et a. (1996) identiied our important states on the base-shear-roo-drit reationship o a propery designed wa. These states are: (1) decompression state; (2) sotening state; (3) yieding state; and (4) aiure state. 2.1.3.1 Decompression state The decompression state identiies the point on the base-shear-roo-drit reationship (Figure 2.3) when the initia stress in the concrete, ci due to post-tensioning and gravity is ost under the action o atera oads. Decompression begins at the extreme tension corner at the base o the wa and indicates the initiation o a gap between the base pane and the oundation. Decompression aso identiies the end o the inear response o the wa. 11
1 aiure state 8 yieding state base-shear (kips) 6 4 sotening state decompression state 2.5 1 1.5 2 2.5 roo-drit (%) Figure 2.3 Base-shear versus roo-drit reationship o Wa WH1M 2.1.3.2 Sotening state The sotening state identiies the point on the base-shear-roo-drit reationship when there is a signiicant reduction in the atera stiness o the wa (Figure 2.3). The reduction in the atera stiness o an unbonded post-tensioned wa usuay occurs in a smooth and continuous manner. Sotening can occur because o three reasons: (1) decrease in the contact ength aong the horizonta joints as a resut o the ormation o gaps; (2) noninear behavior o the concrete in compression; and (3) noninear behavior o the post-tensioning bars in tension. In a propery designed wa, noninear behavior o the post-tensioning bars occurs ater signiicant sotening o the wa due to the ormation o gaps aong the horizonta joints and/or noninear behavior o the concrete in compression. I the initia stress in the concrete, ci is arge, the ormation o the gaps is deayed and noninear behavior o the concrete governs the sotening state. I the initia stress in the concrete is sma, the noninear behavior o the concrete is deayed and ormation o the gaps governs the sotening state (Kurama et a. 1999b). 12
2.1.3.3 Yieding state The yieding state identiies the point on the base-shear-roo-drit reationship when the strain in the post-tensioning bars irst reaches the inear imit strain (i.e., the imit o proportionaity). Prior to the yieding state, the post-tensioning bars behave in a inear-eastic manner. In a propery designed wa, the yieding state is reached ater arge noninear atera dispacements which occur primariy due to the ormation o gaps aong the horizonta joints and noninear behavior o the concrete in compression. Prior to the yieding state, noticeabe damage to the concrete in compression is sma as a resut o the use o spira reinorcement in the base pane. The cover concrete may experience some damage over a sma region near the base o the wa (Kurama et a. 1999b). 2.1.3.4 Faiure state The aiure state identiies the axia-exura aiure o the wa as a resut o crushing o the spira conined concrete. The conined concrete crushes when the spira reinorcement ractures. As described in Kurama et a. (1999b), this is the desired mode o aiure in a propery designed wa. Suicient spira reinorcement is provided in the base pane to ensure that the aiure state is reached at a roo-drit signiicanty arger than the roo-drit at the yieding state. 2.2 Prestressed Beams with Openings The behavior and design o post-tensioned precast concrete beams with openings have been investigated by Abdaa and Kennedy (1995, 1995b, 1995c), Barney et a. (1977), Kennedy and Abdaa (1992), and Kennedy and E-Laithy (1982). These studies indicate that the argest tension stresses around an opening occur near the centerine o the opening on the edges perpendicuar to the post-tensioning orces. Kennedy and E-Laithy (1982) used a truss anaogy to design the beam reinorcement around the openings. A simiar approach is used in this report or the design o the pane reinorcement in unbonded post-tensioned precast was under vertica oads (Chapter 5). 2.3 Monoithic Cast-in-Pace Concrete Was with Openings The seismic behavior and design o monoithic cast-in-pace reinorced concrete was with openings have been investigated by many researchers incuding Ai and Wight (1991), Kato et a. (1995), Kobayashi et a. (1995), Pauay and Priestey (1992), Powers and Waace (1998), Tayor et a. (1998), Waace (1998), and Yanez et a. (1992). Severa dierent methods were proposed or the design o the was incuding: (1) the strength reduction method; and (2) the truss anaogy. In the strength reduction method (Kobayashi et a. 1995), the atera strength o a wa with openings is reduced using an equivaent wa without openings. The truss anaogy method (Tayor et a. 1998) uses struts and ties aong directions o the principe compression and tension stresses in 13
the wa to design horizonta reinorcement to ensure that the atera orces have an adequate oad path to the base o the wa. As the openings become arge, the was begin to behave ike couped shear wa systems. Aktan and Bertero (1987), Park and Pauay (1974), Pauay and Priestey (1992), and Pauay and Santhakumar (1976) are a ew o those who have investigated the seismic behavior and design o couped reinorced concrete shear was. The behavior and design o unbonded post-tensioned precast was with openings is signiicanty dierent than monoithic cast-in-pace concrete was with openings as a resut o the ormation o gaps aong the horizonta joints. Thus, the previous research on monoithic cast-in-pace was with openings and couped shear was is not investigated urther in the report. 2.4 Precast Concrete Was with Openings The atera oad behavior and design o precast concrete was with openings have been investigated by Mackertich and Aswad (1997). A method or designing the was as soid was with an equivaent reduced thickness was proposed. The previous research on precast concrete was with openings has ocused on was that emuate the behavior o monoithic cast-in-pace reinorced concrete was. The use o openings in non-emuative precast was in which gaps orm aong the horizonta joints under atera oads has not been investigated. The research described in this report shows that these gaps have a signiicant eect on the behavior and design o the was. 2.5 Fiber Eement Mode The previous research on unbonded post-tensioned precast concrete was used an anaytica mode based on iber eements as described by Kurama et a. (1996, 1999a, 1999b). This mode, which is reerred to as the iber eement mode, was deveoped using the DRAIN-2DX Program (Prakash et a. 1993). The iber eement mode was used to veriy the inite eement mode described in Chapter 4 o this report. In the previous wa mode, iber eements are used to represent the axia-exura behavior o the concrete wa panes. Each iber eement consists o a number o parae concrete ibers aong the direction o the height o the pane. Each concrete iber has a ocation in the pane cross-section, a cross-sectiona area, and a uniaxia concrete stress-strain reationship (Kurama et a. 1999b). The stress-strain reationship o the concrete ibers is a muti-inear ideaization o the smooth stressstrain reationship or the unconined concrete or the spira conined concrete based on a mode deveoped by Mander et a. (1988). 14
The post-tensioning bars are modeed using truss eements. The post-tensioning o the wa is simuated by initia tensie orces in the truss eements which are equiibrated by compression stresses in the iber eements. The stress-strain reationship o the truss eements is a biinear ideaization o the smooth stress-strain reationship o the post-tensioning stee (Kurama et a. 1999b). In the iber eement mode o an unbonded post-tensioned precast wa, the gaps that orm aong the horizonta joints are represented as distributed tensie deormation in the wa panes. The tensie strength and stiness o the concrete ibers modeing the panes are set to zero and the bonded pane reinorcement (i.e., wire mesh, which is not continuous across the horizonta joints) is negected to aow the dispacements rom the gaps to be distributed in the wa panes. The reduction in the atera stiness o the wa due to the ormation o gaps is represented by the zero stiness o the ibers that go into tension. The iber eements modeing the wa panes are based on the assumption that pane sections remain pane. However, in an unbonded post-tensioned precast wa, the ormation o gaps vioates the assumption that pane sections remain pane in the regions o the wa panes immediatey adjacent to the horizonta joints. Thereore, the oca stresses and strains in the wa panes near the horizonta joints are not accuratey captured in the iber eement mode. However, Kurama et a. (1996, 1999b) determined that the eect o the gaps on the overa atera oad behavior o the was is accuratey captured. The iber eement mode cannot accuratey represent the stress distributions around the openings in unbonded post-tensioned precast was, and thus, cannot be used to investigate the oca behavior and design around the openings in the wa panes. Thus, a inite eement mode is deveoped or was with openings as described in Chapter 4. 2.6 Cosed-Form Soutions or Ininite Eastic Panes with Openings Cosed-orm anaytica soutions are avaiabe (Savin 1961) or the maximum tensie stresses around rectanguar openings in ininite eastic panes or a imited number o oad and opening conigurations. A brie overview o the soutions presented by Savin is given in this section. These soutions are used to veriy the inite eement mode described in Chapter 4. As an exampe, Figure 2.4 shows a inear-eastic pane oaded with a uniorm compression stress, p. The pane is assumed to be ininitey ong in both the horizonta and vertica directions. The presence o the opening resuts in the deveopment o axia stresses (in the horizonta, X-direction) above and beow the opening as shown in Figure 2.4. The stresses near the opening are tensie, with the maximum tension stress, tm occurring at the center o the opening edge. 15