Assume the bending moment acting on DE is twice that acting on AB, i.e Nmm, and is of opposite sign.

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1 232 End cf Do run Assume the bending moment acting on DE is twice that acting on AB, i.e Nmm, and is of opposite sign. n n Distance between BM couple = ^ x 432 = 288 mm o Magnitude of forces of couple = =. 480 N Extreme fibre stress = ^ = ~ A,44 MPc. Stress at node h Stress at node i Consider ndhp: r! J x 4,44 = +2,96 MPa x 4,44 = +1,48 MPa Total force on surface between D and h - x N Position /

2 Position of centroid : 2,96 x 72 x ^ = Nmm 1.48 x = i 280 = 33,6 mm from D., Force on node D x Force on node h x N = +124 N +266 N Consider phiq: Total force on hi ' -r-'-." x 72 = +160 N Position of centroid : 1,48 x 72 x ^ = Nmm 1.48 x = mm from h. Force on h = x (72-32) +88,9 N Force of i = Uftp- x 32 * N +160,0 N Consider ql.j: Total force on ij «+1,48 x = +53>3 N Force on i = +53,3 x * +35,5 N Force on j = + 53,3 x i = +1?.8 N +53,3 N Forces on nodes Node D (204) : +142 N Node h (178) : = +213 N Node i /...

3 234 Node i (152) +71,1 +35, Ncde 3 (126) +17,8-17,8 = 0 Node k (100) Node 1 (74) Node E (48) -107 N -213 N -142 N CA.1.Ll). Possible loads on a small two-storey building with a 126 mm thick concrcte slab were assumed and are shown on Figure 58. n Top of column Bending moment = N mm/n.m width Bending moment couple = P = -!".- 94 M/inm width End fibre stress Stress at node b = -0,655 MPa 2 = ± 1,31 MPa Consider /

4 Consider Abed: Total force on surface Ab = ^ x 72 = -70,74 N Position of centroid (from Case (a)) is 32 mm from A Force on node A = -70,74 x 40-7? Force on node h U -70,74 x 12 Consider bee: Total force ort be -0,655-4-Z2. _ 23,58 N 9 - C. -39,30 N = -31,44 N -70,74 N Force on node b = -<-3,58 x 0,667 = -15,73 N Force of node c = -23,58 x 0,333 = -7,85 N,! N Uniformly distributed load on column Force on nodes 233 and 237 : -2,33 x 36 = -83,9 N Force on nodes 234, 235 and 236 : ,8 N Forccg on nodes on top of column Node A (233) -39,3-83,9 = -123,2 N Node b (234) -31,4-15,7-167,8 = -214,? N Node c (235) -7,8 + 7,8-167,8 = -167,8 N Mode f (236) +31,4 + 15,7-167,8 = -120,7 N Node B (237) + 39,3-83,9 = -44,6 N End of Beam Bending moment = Nmm Distance between bending moment couple Magnitude of forces of couple = = 2 = 288 mm = 125 N Extreme /

5 Extreme fibre stress = * 1>16 MPa Stress on node h = : 7 = +0,773 Mia Stress on node i = Consider ndhp: J - +0,337 MPa Total force on surface Dh C. x 72 = +69,6 N Position of centroid (from Case (a)) is 33,6 in from D. Force on node D +69,6 x = +37,12 N Force on node h = +69,6 x +32,48 N +69,60 N Consider phi.fi: Total force on hi - ^ x 72 = +41,76 N c FJ<^ition of centroid is 32 mm from h. Force on h /...

6 Force on h x (72-32) Force on i - x 32 Iu 237 = +23,20 N +18,56 N +41,76 N Consider (il.j: Total force on ij = +0,387 x +13,93 N Force on i = +13,93 x *7 Force on j = +13,93 x 4 +9,29 N +4,64 N +13,93 N Forces on nodes it) horizontal direction Node D (204) Node h (178) Node i (152) Node i (126) Node k (100) Node 1 (74) Node E (48) Shear stress on DE +37,1 N +32,5 + 23,2» +55,7 N +18,6 + 9,3 = +27,9 N + <1,6-4,6 =0-27,9 N -55,7 N -37,1 N Assuming load on end DE of beam from floor, \all and beam = 208 N Shear stress = 1-- o > = 0,48 MPa I?. Vertical force on nodes 204 and 48 ~ 0,48 x 36 = 17,3 N Vertical force on nodes 178, 152, 126, 100 ond 7': = 34,6 N Vertical Loads on Nodal Points on Top of Beam Distributed load -0,0323 N/mm ' Node 190 Node 199 Node POO -0,032 x 18-0,032 x 36-0,032 x ( ) -0,58 N -1,16 N -1,75 N Node 201 : /

7 238 Node 201 : -0,032 x 72 Node 202 : -0,032 x (?6 + 72) Node 203 : -0,032 x 144 Node 204 : -0,032 x 72 = -2,33 N = -3,49 N = -4,66 N = -2,33 N Total force on Node 204 : -2,33-17,3 = -19,6 N CASE (c) In addition ^o the Loads of Case (b), horizontal loads were applied a3 shown on Figure 59. These loads were arbitrarily selected. The total load on the column from the brickwork was 252 N and this caused a tension of 252 N applied on the end of the beam. B 0,5 MPa 0,778 MPa n O r t U c i ;o t it

8 239 Area on sketch Total load on surface Moments about node Rec tangle Triangle Position of centroid (mm) Bvwr 81, ,2 from r rwxs 48, ,4 from 3 sxyt 55, ,8 from t tyzu 32, ,7 from u uzcm 34, ,8 from c Horizontal forces on nodes Node D (237) Node r (232) Node s (227) Node t (222) Node v (213) Node C (198) -39,2-42,4-23,6-24,4-27,0-28,8-15,7-16,3-17,1-17,5-39,2 N = -66,0 N = -51,4 N = -44,5 N = -33,4 N = -17,5 N -252,0 N End of Beam It was arbitrarily decided to distribute the tension on the end of the beam so that the stress at D was twice the stress at E i.e. 0,778 and MPa respectively. Check : x ^32 = 252 N Nodal Point Stress on Node Nodal Point Stress on Node D 0,778 k 0,519 h 0, ,454 i 0,648 E 0,389 J 0,583

9 240 Area on sketca Total load on surface Moments about node Rectangle Triangle Position of centroid (mm) ndhp 53, 'G 35,5 from h phiq 49, ,4 from i qi,1 44, ,4 from j,jkk 39, ,3 from k kkll 35, ,2 from 1 11EE 30, ,1 from E 252,0 H ori zontnl forcer. on end o f benni nodr':; Node D (204) +26,5 = +26,5 N + 37,1 = +63,6 N Node h (178) + 27,2 +24,1 = +51,3 N +55,7 = +107,0 N Node i (152) + 24,9 f 21,8 = + (';6,7 N +27,9 = +74,6 N Node j (126) +22,5 + 19, ,0 N 0 = +42,0 N Node k (100) + 20,2 + 17,1 = +37,3 N -27,9 3 +9,4 N Node 1 (74) +17,9 + 14,8 = +32,7 N -55,7 r -23,0 N Node E (48) + 15,5 = +15,5 N -37,1 = -21,6 N O CASE (c) The loading was the same as in Case (c) except that the horizontal stresses on the stub column and the horizontal stresses on the end of the beai.i were 10 times higher. ilor i zont.-i ] f'orr.r ", on column edge nodes Node B (237) : -392 Node r (232) : -424 Node s (227) : -244 Node t. (222) : -288 = -392 N 236 = -660 N 270 = -514 N 157 = -445 K

10 241 Node v (213) Nodo C (198) n -175 N N Horizontal forces on end of beam nodes Node D (204) N +37 = +302 N Node h (178) = +513 N +56 = +569 N Node i (152) a +467 N +28 = +495 N Node j (126) = +420 N 0 = +420 N Node k (100) = N N Node 1 (74) = -*-327 N -56 = +271 N Node E (48) +155 = N -37 = +118 N N NOTE: The element numbers higher than 173 and the nodal point numbers higher than 185 change in the case of the cracked junctions, i.e. Uncracked Cracked Node B Node r Node s Node t 2?2 224 Node v Node C Mode A Node b Node c Node f ctr*.

11

12 Author Boardman Vivian Reginald Name of thesis The Effect Of Compressive Stress On The Expansion Of Brickwork And Its Implication In Buildings PUBLISHER: University of the Witwatersrand, Johannesburg 2013 LEGAL NOTICES: Copyright Notice: All materials on the University of the Witwatersrand, Johannesburg Library website are protected by South African copyright law and may not be distributed, transmitted, displayed, or otherwise published in any format, without the prior written permission of the copyright owner. Disclaimer and Terms of Use: Provided that you maintain all copyright and other notices contained therein, you may download material (one machine readable copy and one print copy per page) for your personal and/or educational non-commercial use only. The University of the Witwatersrand, Johannesburg, is not responsible for any errors or omissions and excludes any and all liability for any errors in or omissions from the information on the Library website.

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