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APPENDIX A Frame Analysis
APPENDIX A FRAME ANALYSES DETERMINATION OF DEFLECTIONS The procedure employed i n computing deflections was that suggested by Vanderbilt, Sozen, and Siess.^"'"^^ I n t h i s method, the structure i s divided i n t o beam and plate elements. Deflection at the center of a panel i s taken as the sum of deflections of the i n d i v i d u a l elements. The stnacture i s separated i n t o parts as shown i n Fig. A-1. These parts represent, approximately, the deformed shape of a slab. Panel dimensions are designated L f o r the long span and S f o r the short span. The i n t e r i o r of the panel i s a simply supported plate 0.6L long and 0.6s wide. Deflection at the center of the plate i s computed f o r a uniform load by the well known formula given elsewhere.^^^^ An equivalent frame i s taken i n the d i r e c t i o n of the long span of the panel. The frame i s made up of a slab s t r i p and supporting columns. Width of the slab s t r i p i s taken as 0.2S each side of the column l i n e . Deflection of the frame midway between colimins i s computed using mo- ment diagrams obtained by moment d i s t r i b u t i o n . ^ ^ ^ ^ From the longitudinal centerline of the frame to the edge of the frame, the slab i s assumed to act as a cantilever beam r i g i d l y attached at the frame centerline. Deflections at the end of the cantilever are computed taking i n t o account computed r o t a t i o n of the frame centerline. Total deflection at the center of a loaded panel i s taken as the sum obtained from the three parts: the p l a t e , the frame,and the cantilever beam. A-l
Three separate frames were used i n computing deflections i n the loaded' panels. Locations of these frames are indicated on the roof plan i n Fig. A-2. Frames 2 and 3 were.used t o obtain deflections midway between colianns of the loaded panels. Stiffness of the structure was the same along column Lines C ajid D. Therefore Frame 2 could be used t o obtain deflections f o r a l l three tests. Frame 3 was used only t o determine deflections along column Line E during Test I I . Because of the edge beam along Line E, s t i f f n e s s of Frame 3 was s i g n i f i c a n t l y d i f f e r e n t from that of Frame 2. Frame 1 was used t o calculate rotations at the ends of the loaded spans. Effects of these rotations were combined with deflections at the ends of the cantilever beams. Contribution of the frame to panel deflection required the use of two frames f o r both Test I and Test I I . I n Test I , the frame along column Line C was loaded from both sides. Along column Line D, the frame sup- ported load from only one side. The average of deflections computed f o r each frame was then used t o obtain the contribution at the center of the panel. I n Test I I , the average deflection from two frames was again required. I n t h i s case, however, the frames were of d i f f e r e n t stiffnesses but subjected t o the same loading. Frame 1 was used f o r each of the three tests. This frame represented the structure along colvmin Lines 3* ^) 5} and 6. Fig. A-3 shows d e t a i l s of the frames analyzed and the loads that were applied. A l l coltmins were considered f i x e d at the foundation. At B i n Frame 1, a hinge was assumed to be present i n the slab f o r Tests I and I I I . The assumed hinge was located at the point where depth of the slab changed from 2h i n . to 8 i n . Where the colimin and mezzanine intersected, a hinge was also assumed. The influence of these assumed hinges on computed values was checked and found to be small. In Test I I , the slab was considered f i x e d at B. Influence of t h i s assumed f i x i t y was also small since the loaded span was quite distant. However, convergence i n the moment d i s t r i b u t i o n procedure was hastened by t h i s A-2
assumption. For sim i l a r reasons, the frame was assumed fixed a t 3 i n Frame 2 for Test I I I . The outside spans of Frame 2 were assumed fixed a t t h e i r ends. At these locations the slab was supported by a 12-in. t h i c k wall. Contact with s o i l outside the building as we l l as a short e f f e c t i v e length created by the mezzanine gave the wa l l considerable r i g i d i t y . Conseqiiently, the assumption of f u l l f i x i t y appears reasonable. At l i n e s 2 and 5 of Frame 3, the slab framed into the wa l l . The assump- tion of f i x i t y at 2 and 5 was made to simplify computations. This assumption should not greatly influence the computed r e s u l t s . I t i s seen i n Fi g . A-3 that load consists of a uniformly distributed portion with a triangle or trapezoid load superposed near the middle of each span. Load was determined according to the trib u t a r y areas outlined i n Fi g . A-h. The uniform load was that supported over the portion of the equivalent frame that was loaded. The triangular or trapezoidal loads were those from the panel areas supported by the frame. The angle aÌ ^ i s a function of the span lengths of the panel. COLUMN LOADS The equivalent frame(â¢^-^'^^^ consisted of the roof slab and supporting colimins. For i n t e r i o r spans, the frame was one panel wide, extending one-half panel each side of the column centerline. Along the edge of the building, the frame was only one-half panel wide. Although the half panel extended to one side of the columns i n the actual structure, e f f e c t s of la c k of symmetry were ignored i n the ana l y s i s of the equivalent frame. Five frames were used to determine colvmin reactions for the three t e s t s . Frame locations are indicated i n Fi g . A-5. The frames were taken i n both p r i n c i p a l directions. I t was expected that analysis of frames i n orthogonal directions would y i e l d s l i g h t l y d i f f e r e n t r e s u l t s . In calculation of s t i f f n e s s e s , a l l sections were assimed' uncracked. A-3
Each frame and the loading used i n i t s a n a l y s i s i s shown i n F i g . A-6. Coliatms were considered fixed a t the foundation. Two di f f e r e n t end conditions were investigated for Frame k. In Frame ka., colvunns a t B and A-B were asstmied hinged a t the mezzanine l e v e l and the beam was assumed fixed at A. In Frame kt, a hinge was assumed i n the beam at B. Both frames were analyzed for Test I loading to determine e f f e c t of these asstmiptions on computed column loads. End moments were obtained by moment d i s t r i b u t i o n . ( ^ ^ ^ Using these moments and span loading, coltmin reactions were obtained. A.4
VJ1 FIG. A-1 DEFORMED SHAPE OF A SLAB
> FRAME FRAME 3 FIG. A-2 FRAMES USED IN COMPUTING DEFLECTIONS FRAME 2
> --3 B A-B W W \W FRAME l -TEST I irrrfTTTrn E m D 777 m FRAME l-TEST 31 B E 777 [TTTTTrmi /77 m FRAME I-TEST m B A.B 3 m m FRAME 3-TEST 31 21 3 I 4 1 5 m m m rn FRAME 2 - T E S T I 6 777 fn fn m m iiT FRAME 2-TEST I T ftr rfi rh FRAME 2-TEST I H FTG. A-3 FRAMl'Ì S /JTO LOADS TN .";''MPrTTNG DEFTF.CTIONS
0.2L I 0.2L > CD S 1 0.2L 0.2 L 0.2 S 0.2 S 0.2 S 0.2S FIG. A-k T1TSTRTBUTI0N OF PANEL LOAD TO FRAME t t
3 FRAME 4 FRAMEC FRAME E FRAME 3 F R A M E 5 YIC. A-5 imi'.S u-SrJ IN COriPUlINa COLUMN LOADS
E m TtT > itT D m c m e d B A-B FRAME 3 TEST H TfT B A-B A FRAME 4 a - T E S T 1 B A-B A /77 m FRAME 4b-TEST I B A-B 2 l l l l l l l l . i l 3 777 FRAME C - T E S T I 6 FRAME C - T E S T H I i - L i M i n ^ I /77 ;77 FRAME E - T E S T H 777 TV FRAME 5-TEST UL FrG.A-6 FRAMES AND LOADS IN COMPUTING COLUMN LOADS
