W_. “W'— 107 100 THS ALLOWABLE BEAM DEFLECTION AS LIMITED BY PLASTER STRAIN Thesis for the Degree of M. S. MICHSGAN STATE COLLEGE FEorence Heien Dyer 19550 o‘t‘ :. ' I‘_-_’1‘;~‘- LiL'X‘ ', ‘.,.',‘_|.‘ H I_‘, -.' p . . I II ‘VV.’ 3 . y - I I. ‘I. 5.3 ‘ .. a! C‘ ‘ . . 4 u " -‘. ' "I ‘X. ~Jfl ' .v th'}, ' . d I. 'y I 7' ' q 2, M. N- t, " . . ’.-- ‘7, “MM J57}: ~’ “NH; "-3.4th ’tx‘J'JK‘} '4" ‘ ’~ ‘ a n9 czfn‘i ‘ \- I h v _ ,1 a p n . \‘ " r» ‘o A.” “IS-1’ R ‘hé -$ 4 “N" '13 ‘\ ’2‘ I} “ ‘£ ' A o . '.' ‘, 4' V " f ‘ \ . I '6 J ltqfita 51.53.,‘(17‘ 3". .‘Z‘. Ll“ ‘ , . v. ‘s. ‘2 M‘\ pk)" xi“ ”3 l‘ _ a\. .. . f .57; t if Mr W {Id «x '! ‘ . . wag.‘¢tfikamw,: , 5 ~ WW: 2- .9 v ». . I /4' 3'1”“ K1“ 5y» .\ I I... ’ Y (\Q’ “ f' I ll 4.? ‘_,a- 7‘ 7. ' .‘ -'- , ‘ ‘ . l " ‘9‘ ‘ ' ’ ‘. . e g' _ ." - v : - u 1 ‘s ,k . ,‘. l . V. _ \ , Iv, ,.' I . u ) , u I‘ . ‘ . lukw’.’ ' ‘ ' P 2"". “‘w .H. .‘7 ‘2’," "‘5 " . ‘ f - _. .. ' - . t ,d, t .' ‘\ ‘3 i.) 3 2v" ‘4‘ 1 - ‘ .:' . ‘. W . .‘ I . I t n - - 1 . , ~ I _ V '..’ . . _\ ' ‘ o I\ ‘ , a ‘- A 1': u '1”. ‘w‘ 1‘ 5 034$: W" .A-» V' 4‘ Ll.“." . L).__ _' I i k F L . 8 I‘ . . lz'Wl ’5 f This is to certify that the d ' ’a thesis entitled ALLOWABLE BEAM DEFLECTION AS r-—-_ w—w ‘m‘r '4 _—‘, LIMITED BY PLASTER STRAIN presented by FLORENCE DYER has been accepted towards fulfillment of the requirements for M. 8. degree in CIVIL ENGINEERING .I " .\ U o ' O M I“ 2‘ ‘ U ' v ' ' .u . . . Major professor I May 25, 1950 -Y ' . Date 5.! ._ a—vw‘ vv-Ms c" 'J “m ‘,v-‘ I "‘4- :1: *1‘5'3;T€~";~*wm2'-.. .' ., . """~n' u‘w I : -' ”u, I .r v . fl, <.-‘.‘ .‘I'f'fgvl.’ .5 ' 'YA.‘\ , " <~_ ."_,‘.‘r"L-‘ 4" ink). . V‘s.» " ...- .39 s ‘ 0. ,_. A ..c U ‘- ' ‘.- . Q ’ - 4" '. ‘ ~‘\ .7: ‘t-- —. p . 353v «”5: Ma. . Y: L ALLOWABLE BEAM DEFLECTION AS LIMITED BY PLASTER STRAIN By Florence Helen Dyer A THESIS Subndtted to the School of Graduate Studies of Michigan State College of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of HASTER OF SCIENCE Department of Civil Engineering 1950 ACKNOWLEDGMENT The writer wishes to express her appreciation to Doctor Charles 0. Harris, Professor of Civil Engineering, for his guidance and help in the development of this dissertation. *********# *******s *ttttt sass ** * $337 '73 TABLE OF CONTENTS Page INrmDTJCTIONOOOOOOOOOOOOOO.0.00.0000...OOOOOOOOOOOOOOOOO. 1 THEORYoooooooooooosoooooo0000000000000...cocoon-000000000 3 EXPERIMENT............................................... 14 DISCUSSION............................................... 20 CONCLUSIONS.............................................. 25 REFERENCES...o........................................... 26 APPm‘TDIXOOOOOOOO00.00.0000...00......OOOOOOOOOOOOOOOOOOOO 27 INTRODUCTION The object of this thesis is to determine a criterion for the calculation of the maximum value of beam deflection due to live load which will not cause the plaster strain in tension, in an associated ceiling or wall panel, to exceed a maximum value beyond which plaster failure would result. In the fifth edition of the Steel Construction Manual of AISC (3) is the existing specification: "Beams and girders supporting plastered ceilings shall if practicable be so proportioned that the maximum live load deflection will not exceed 1/360 of the span." It seems that this blanket rule of thumb might be refined for designs of greater precision if the value for the limiting strain were deter- mined for various plasters; and if the relation between strain and deflection were stated mathematically for representative conditions of loading. As far as could be determined by reference to the Industrial Arts Index and the Engineering Index no articles are available on plaster strain caused by beam deflection. Plaster The plasters used were those packaged by the U. S. Gypsum.Company. Several types and mixtures were tested; basically they all contain calcined gypsum. Plaster of paris is the pure form (2CaSO4 plus HZO). Keenes cement, a hard finish plaster, is made by ndxing alum with the calcined gypsum and recalcining. -1... The finishing lime plaster is made with hydrated lime, Ca(OH)2, slaked and mixed with guaging plaster. In this case the development of strength is a progressive process depending on the formation of CaCO3 by 002 in the air." THEORY Plastered Beam The maximm deflection (3) in a beam of length '1', under uniform loading is: beef 38 where ‘5'13 the deflection; 'w' is the load per unit of length; “8' is the modulus of elasticity, a constant for the material of the beam within the proportional limit; and 'I' is the moment of inertia about the horizontal axis ’of symmetry of the section called the neutral axis. The moment of inertia (4) in Figure l is: (2) I 2 bd3 '1'2 The maximum stress (4) in a homogeneous beam, as in Figure l, is: (3) f : Me , the flexure formla, I where 'q" is the unit stress; 'M' is the mximum moment; and 'c' is the distance from the neutral axis to the extreme fiber. The amount of stress in any fiber is proportional to the distance from the neutral axis as shown by the arrows in Figure 2. Therefore the maximum stress occurs in the extreme'fiber. By Hookes law (4), strain is directly proportional to stress: (4) E:_u_;_ or egg; 6: E -3- or combined with equation (3) (5) €max : lunxc BI Thus the maximum strain denoted by‘e'also occurs in the extreme fiber. In the case of) a plastered beam as shown in Figure 3, it is necessary to consider the two materials acting together as an equivalent homo- geneous section (Figure 4), where the ratio of the widths bl/b is dir- ectly proportional to the ratio of the modulus of elasticity of plaster to the modulus of elasticity of the beam; the thickness 't' of the plaster remains unchanged. So, b1 I Ep or bl = E ; b b EB E6 ' Let : " th b : b g b en 1 g . EB It will be shown that the value of 'I' of the transformed section in Figure 4, differs from the value of 'I' of the beam section in Figure l, but that the difference is very small and may be disregarded. Also, an exact value for the distance from the neutral axis to the ex- treme fiber '§" will be found. However, the value of '5." differs so slightly from the value of d/2 + t (the 'c' distance in Figure 3.) that this difference may also be disregarded. The distance from the neutral axis to the extreme fiber of the transformed section in Figure 4, is found by balancing the moments of the areas about the neutral axis: MQ+t-§) : bfiw'f) 2 2 bd2 y bdt - bdy - blty - bltz 2 "2 ' on+bmy =g£+be+ 2 b 113.2 2 y : §§E_ + bdt + bgt? blt + bd substitute: b =bg , where g: E2 E on t = dk , where k = t/h § g bd2 + bkd2 t bgkzdz '5 "'2 " bgkd 1‘ bd .. 2 y- gu+2k+s) 2 . legk The term gk2 in the numerator may be neglected since it is very small compared to the other terms, so: §=dfl(1+%@1o~: lept BB (1 This is the distance from the neutral axis of the transformed section to the extreme fiber, that is: and neglecting the term t/d Ep in the demonimator E B -5- (6) ’czdft z The value of t/d lip will be very small. The modulus of elasticity of plaster will nearlyEglways be much smaller than the modulus of elas- ticity of the beam material. I? would be a large value for t/d occur— ring when 1/2" of plaster is applied to a 5" beam. The moment of inertia of the transformed section about its centroi- dal axis is found by using the parallel axis theorem (4). HI ll .. 2 3 s - kd 2 bd3 + bd Ei/Z {- kd - y] + b k d + bgkd[y __1 12" 12 2 very small 2 2 bd e bd\ 0.003 \§\ \§\ 0.... Ratio of bese.depth to been length Figure 5 Relation of deflection-length to depth-length ratio for s range of t/d values. Plaster strain taken as: 0.0000 .9- letie or been deflection to been length Plastered wall Panel When a plastered wall panel is supported as in Figure 6, the de- flection of the supporting beam at 'A' results in distortion of the plastered wall panel. It is assumed that the support at 'B' in Figure 6 does not deflect. The relation between the deflection of the beam and the unit strain in the plaster is found as follows: The unit strain, 0' , in the plaster is (12) 5: 8‘ Where ‘8' is the deflection in the beam of 'A' in Figure 6 and 'L' is the length of the plaster panel. The shearing modulus of elasticity, 'G', is the ratio of unit shearing stress 'Ss' to unit strain, K . (13) G: s ,or f- The panel is in a state of pure shear so the principle stresses are: (14) s0 = st = s as shown by the stresses on elemental areas in Figure 7 and Figure B, or the maximum unit strain is, from Hookes law, (15) = s [- usc = s (1+v)=GS(1n e t E ‘L’Fz‘fl F's—2 Where '3' is the modulus of elasticity in tension and‘fl'is Poissons ratio. Since -10- Figure 9% Q“ 3s 43% 5s 1513“” 6 1'15“” 7 -l'l' (16) E 3 2(1 4-4) G. it follows that, (17) 6 = 5/2L 01' _§_=2e. L EXPERINENT The plaster test bars were made up in a form.as shown in Figure 9. The nominal size of each bar was 1“ x 3" x 24". Three sets of forms were used so that three similar bars could be made at one time. Nine different mixtures of three bars each were made. The pro- portioning of the mixtures was that recommended by the Gypsum Com- pany (5). As soon as the plaster had set enough to be sufficiently firm the bars were removed from the forms and stored so that air could circulate freely on all sides for uniform drying. Small holes on the surface of the bars were patched with plaster of paris to reduce the concentration of stress at thoeapoints. When the bars were thoroughly dry, usually about one week, after forming, they were submitted to a bending test on the apparatus shown in Figure 10. The supports at 'A' and ”A'" were placed 8' apart, or approximately at the third points of the bar. The load applied at 'B' was divided evenly between 'C' and 'C". With this arrangement the bar was subjected to a constant moment between the two supports as shown in the diagram in Figure 11. It is believed (1) that the break- ing stress in a test of this kind on brittle material is less than center loading due to the break always occurring at a weaker section. The weight of the bar was so small in comparison with the applied loads that the weight was neglected in the calculations of stress and strain. Loading Disgra- ulfi loeent Diagram Figure 11 -15- In the testing machine the loads were applied gradually and read- ings of deflection were made by means of diai guages, 'D’ and 'D" in Figure 10. From.these values and the careful measurement of dimensions of each bar, values were calculated for the modulus of elasticity, maximum.tensile stress.and maximum strain in each bar. Average values of these quantities are tabulated in Table I, and the stress strain curves are shown in Figure 12. The modulus of elasticity was calculated as follows: The ex— pression for maximum deflection under third point loading is; Jmex: 23P13 or §_5__vn__§_ 648 El 1296EI Where P is the load at one end and W’is the total load applied at'B'in.Figure 10. From that equation: E = 23m") I1296 maxf_f The value of the length, '1', was 23.82 Smax was the sum of the two guage readings for any one load. The moment of inertia, 'I', was calculatedfor each bar using: I = bd3 12 where 'b' was the width of the bar,.nominally 3", and 'd' is the height of the bar, nomdnally 1". There were minor differences between the dimensions of different bars due to variable shrinkage in the plaster and perhaps some swelling of the wood in the forms. -15- l 1 «9“» 000 9’9:st try: as 000 'P ‘de ‘fiL‘.£& 3 9* ~ y’v‘ § 5 3'00 5 / 53 200 / , :5 ‘.r-,¢s11°' 3 MW“ “"1 . 0.0002 0.0004 0.0000 unit strain Figure 12 emanate stress-strain rela tests outplastee bars -17- tionlhips fro. bonding 0.0000 TABLE I Nflx Average Average .Average Average Average specific E x 10b Max. max. Max. Wtefi/in3 (tension) in Cement plaster neat 0.0542 1.37 513 0.0435 0.000375 Cement plaster sand 1:2 by wt 0.0653 1.58 625 0.0446 0.000395 Finishing lime plaster 0.0373 0.33 118 0.0417 0.00359 Cement plaster vermiculite 1:1 by vol. 0.0325 0.37 129 0.0711 0.00350 Average for graph Figure 13 0.0037 WOod fibered plaster neat 0.0511 1.25 612 0.0542 0.000498 Keenes cement hard 0.0580 1.40 827 0.0637 0.000590 Average for graph Figure 15 0.000544 Plaster of paris 0.0475 1.02 777 0.0887 0.000760 Guaging plaster 0.0528 1.19 885 0.0868 0.000728 Moulding plaster 0.0479 1.24 905 0.0830 0.000731 Average for graph Figure 13 0.00074 -18.. Tensile stress was calculated from the flexure formula, I 3 Pic , I where 'ii' was the mimm moment due to the given load, "17': .11 3 V! x 7.9 or M 3 W 0 3.95 2 c = d/2 and 'I' is the same value as before a constant for each bar. The value of strain was the ratio of the stress to the modulus of elasticity. 6:}; E DISCUSSION The family of curves in Figure 5 represents the variation in the limiting allowable ratio of beam deflection to beam length, 3/1, as the depth to length ratio of the beam, d/l, changes when the plaster strain is assumedto be 0.0005. The third variable is the ratio of plaster thickness to the beam depth, t/d. Each of the five curves represents a different value of that ratio. The dashed line is, 6/1 I _1_ , or the specified maximum value of 5/1. The grigi: in Figure 5 shows that the depth to length ratio is equally as important as the deflection to length ratio, because as the 'd/l' ratio increasés the limiting allowable ' 5/1' decreases. From the herding tests conducted on the plaster bars the average naxinmm values of strain were calculated (see Table I). Using these values the three curves in Figure 13 were plotted. The t/d ratio is taken as 0.06 because that is the average t/d ratio used previously and the variation between the different values of t/d was snall (Figure 5). The values of émax are those found experi- mentally. 0f the nine types of plaster tested there seemed to be three groupings for 6, shown averaged in Table I. These values were low compared with a value of 0.0013 for plaster of paris in Properties of Engineering Material, (2). It is evident from these two graphs Figure 5 and Figure 13, that the limiting value of ' 8/1' is dependent upon the value of d/l and the value of mximum plaster strain; and is affected, though in a lesser degree, by the thickness of the plaster. -20- é ..-\ \ \ \ \ .° .3. Ratio of been deflection to been length 0.08 \ c. 0.000,. \N latio of bean depth to been length 18 Relation of deflection-length to depth-length ratio lights the asperieental values of plaster strain. t/d taken as: 0.00. 0.10 In equation (9) where €max ‘5 Sm §x24 (1+2 t/d) 5 it is seen that the maxim allowable deflection is inversely propor- tional to the maximum value of plaster strain and to the depth of the beam and directly proportional to the square of the length; thus, II N H M 8m 6:1an where k "5111 y. 2 why The mxinnlm allowable deflection to length ratio, J/l, my be found by using the graph in Figure 14, if the value of maximum plaster strain is known and the depth to length ratio, d/l, of the beam may be calculated. The value for 'd/l' is found on the abscissa and its ordinate is followed until it meets the curve of the desired plaster strain. The vertical coordinate of this point is the naximum allow- able deflection to length ratio for that beam. Comparing the “/1 values found by using equation (9) with the allowable naximum ratio of for average values of plaster strain .3. 360 it is seen that ' 5 -'-' 1 ' exceeds the allowable maximm of T 3% equation (9) as values of d/l increase; though for the lower values of d/l the value of A71 might be considerably greater than :5.1: T350 Though a 5/1 of 1 has been considered "safe", actually the maximm live load deflection for which the beam is designed my -22.. letio of been deflection to been length \ \€,o \ ~ 0.003 \ \ \ 200,100 \ \sow\ 6; \ 0.%’s \ 0.002 V 6“ O W \. ///// es 0.001 \ x— 1 0.03 0.04 0.05 0.00 0.07 0.08 0.09 0.10 Ratio of been depth to been length Figure 14 neletion of deflection-length to depth-length ratio for e reuse of pleeter strain veluee. t/d taken us 0.06. never be attained; also the action of the lath, especially metal lath may strengthen the beam.and so reduce deflection; and in the plastered ceiling, failure in the plaster may produce a fine crack or series of cracks which does not constitute failure of the ceiling because they are so small. Equation (17), Smax :ZémaL shows that in the case of a plaster panel the actual amount of the deflection of the supporting beam.is critical when the ratio of the deflection to the length of the plastered panel equals twice the maximum.allonable strain in the plaster. COIICLTTSI ON S The foregoing tests and discussion seem to support the followirg conclusions: (1) The limiting value of deflection in a beam under a uniform (2) live load which supports a plastered ceiling is: 2 Sum "" k .3...— émx‘ d Where '1' is the length of the beam; 'd' is the depth of the beam, emx is the maximum unit strain in the plastered ceiling; and 'k' I 5 orQ186 when t/d is A 2M1 1» 2 t/d) taken as .06 . The limiting value of deflection in beam which supports on end of a plaster panel at its midpoint when the other sup- port of the panel is fixed is: Smx: ZémL Where 'L' is the length of the panel, ' Gm' is the maximm unit strain in the plastered panel. (1) (2) (3) (4) (5) REFERENCES Davis, Troxell and Wiskocil, "Testing and Inspection of Engi- neering materials" MoGrawbHill Book Company (1941) Nhrphy, G. "Properties of Engineering materials" International Text-book Company, (1947) Steel Construction flannel of the AISC, (1947) Timoshenko, 3. “Strength of materials" Part I D. Van Nostrand (1940) United States Gypsum Company "Directory of Building Materials" (1949) ' 1.0 9.9 10.7 y. I j: I ' ;;< .® . JD 3 d .O.’ u b J9 4 Cement plaster neat 16 15.5 18.5 t ' - - . f Os-et plaster sith vermiculite 3K? L Plaster ct peris 15.9 11.8 10.4 4 a 4‘ L———————L‘ h———————d .____Z___ Celent plaster with sand 10.1 10.7 0 c, Iced-fiber“ plaster 15.6 12 .l 0. . - J 4- D cl 9 O i. 12.? I008 L_____J chasing plaster Right side of bar is tensile side. 15.4 16.4 12.3 Oi. I1 0 — l Finishing line plaster 8.1 11.4 10.2 18.7 8.4 oh 8.4 N leuldins Plaster lumber above each cross-section is the distance of the break free the end of the bar in inches (distance tron right-hand end facing bar in test). lunbers on the feces of the cross-sections represent the aproxinate values of the depths of the stress-raisers outlined nearby. Stress-raisers in the compression side eere not outlined. APPEIDIX Figure 15 -27- DOM USE ONLY 0 O . , Q.“ . x. i8. 1 v i F. 11.5 tastiest 5.1.3.1113”! I ri p I n \i I; .\.-a t I 0 1 II. n \u \h J l \t . 1.! l l , - - \ \\ \\ ‘ k ' [Lil [I i ll! I.’ . .n . _ 0.. . t l . '. . Ca " .2 .e _ .. 0.. . w.” P ll‘.‘ f Jill W'll‘llf'llflli'll'llflfl’ 3 1 7 0 3 O 3 9 2 1 “3 W" I W