CE113 Structural Laboratory Reinforced Concrete Beam Assignment #3 Reinforced Concrete Beam Pretest Predictions

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1 CE113 Structural Labratry Reinfrced Cncrete Beam Assignment #3 Reinfrced Cncrete Beam Pretest Predictins Assignment #3 is an individual assignment t be submitted in draft frm the week f the cncrete cylinder experiment and in final frm the week f the beam test. Fr draft calculatins, use values f assumed f c and Ec as stated in lab. The yield strength will be given by the instructr based n rebar tests (nt perfrmed by the student). 1. Assignment #3 requires students t calculate an estimate f the lads crrespnding t majr stages f beam respnse and t prepare a summary f the predicted failure lad, mde and lcatin fr their beam. Deflectins may als be (ptinally) calculated. 2. The significant stages f respnse and crrespnding lad include: 1) Initial flexural cracking, Pcr; 2) Steel yielding, Py; 3) Cncrete crushing due t flexure after steel yielding, Pm; and 4) Shear failure (befre r after steel yielding), Pv. 3. Assuming flexural failure, calculate the mment and lads assciated with the 3 stages f flexural respnse: Cracking f the cncrete due t flexural tensin: Cracking mment, M cr and crrespnding cracking lad, P cr. Fr the mdulus f rupture, fr, test data shw that fr ranges between 8(sqrt(f c)) t 10(sqrt(f c)). Use an apprpriate value, as discussed in lab. The flexure equatin can be applied and used t calculate Pcr. Nte that the stress is the mdulus f rupture, the mment f inertia is the uncracked value and the lcatin f cracking is the extreme bttm fiber f the beam. Frm this equatin, the flexural strength f the beam at first flexural cracking, M cr, can be fund. y is the distance frm the NA t the bttm f the beam where the tensile stress is maximum. c tr is the distance frm the extreme cmpressin fiber (tp f beam) t the NA. 1

2 Ignring the beam self weight and setting the applied mment equal t the flexural cracking strength f the beam, ne can slve fr the ne unknwn, P cr. Since M cr defines the mment at which the cncrete cracks in flexure, P cr represents the applied lad that causes initial flexural cracking. The accuracy f P cr is especially sensitive t f r. Tensile yielding f the steel reinfrcement: Yield mment, M y, and crrespnding yield lad, P y. Use the value f rebar yield strength given in class and initially assume that the cncrete stress is still linear s that the flexure equatin can be used. This equatin applies t the cracked transfrmed sectin, allwing the calculatin f the Mment when the steel just reaches yield, M y. Rearrange this equatin t slve fr M y. The yield lad, P y, can be calculated by setting the applied mment equal t the yield mment. 2

3 Because the flexure equatin assumes a linear cncrete stress distributin, the maximum cncrete stress shuld be checked. Cncrete stresses stay linear t abut 0.5f'c, but the equatin can be used withut t much errr up t abut 0.7f'c. This equatin allws calculatin f the maximum cmpressive stress in the beam. If this stress exceeds 0.7f'c fr yur beam, then see the instructr fr a mre accurate (and invlved) calculatin f M y. At flexural failure after steel yielding, the cncrete crushes. The ultimate mment at flexural capacity, M ult and ultimate lad, P ult, can be calculated using a prcess smewhat different frm that fr the ther stages. M ult is larger than M y due t: 1) the mment arm slightly increasing, and 2) strain hardening f the steel. Crushing f the cncrete nrmally ccurs prir t fracture f the steel, but either cnditin is assciated with very ductile beam respnse and significant cracking and deflectin. Determine the ultimate mment using the fllwing equatins: At ultimate, the cncrete in tensin has cracked and the cncrete in cmpressin has stresses that vary in a manner similar t the stress strain curve we btained frm the cylinder test. T make the calculatins easier, the actual nnlinear cncrete stress distributin is assumed t be cnstant ver a smaller depth, identified as the distance "a". Since steel can strain harden the maximum stress in the steel at this cnditin may significantly exceed the yield stress, f y. If yur beam has nly ne rebar, then yu may assume a 25% increase beynd the yield stress, and thus take the maximum stress in the steel when the cncrete crushes as f ult = 1.25f y. Call this stress f ult even thugh it is much less than the fracture strength f the rebar. If yur beam has mre than ne rebar, then simply use f ult = f y. 3

4 The ultimate mment capacity (r flexural strength) f the beam is determined by the cuple cnsisting f the tw frces C and T. Fr equilibrium, T=C. T is the frce in the steel and can be used t calculate the mment capacity, Mult=T x mment arm (this is simply the tensile frce times the mment arm). The expressin fr the distance "a" is determined frm C=T, slving fr "a", as shwn in given equatin and explained in lab. The lad at flexural failure (r ultimate), P ult, can be calculated by setting the applied mment equal t the ultimate mment. The beam may als fail in shear, r diagnal tensin, if the shear capacity f the beam is less than the flexural capacity. The average shear stress at which a diagnal tensin crack frms depends n many factrs, including the presence f flexure. There is cnsiderable cmplicatin in using a "shear stress" t estimate the shear strength f a cncrete beam. The value f shear stress, v cr, at initial diagnal tensin cracking may be estimated n the range f 2(sqrt f c) t 3.5(sqrt f c). In regins f high shear and mment, flexural cracks will frm first, reducing the average shear stress at diagnal cracking t apprximately v cr =2(sqrt f c). In regins f high shear and small mment, then "web shear" cracks may frm at a stress level given by v cr =3.5(sqrt f c). Hwever, an alternative research based equatin that accunts fr arching actin, d/a rati (i.e., Vd/M), depth, and flexural reinfrcement (but nt ptential aggregate size effects) can be used fr average shear stress: v cr (psi)=57.1(rh*f c) 1/3, where the flexural reinfrcement rati, rh=as/(bd) and f c is in psi. The shear strength f cncrete, V c, may then be taken as: V c =(v cr )bd. This calculatin will be discussed in lab. By setting the shear strength f the cncrete beam equal t the applied shear (i.e., V c = P V /2), P V can be calculated (P V =2V c ). This represents an estimate f the lad that wuld cause a shear failure in 4

5 the cncrete beam. This failure will ccur nly if anther mde f failure has nt ccurred prir t reaching this lad. Nte that many rutine reinfrced cncrete beam applicatins use stirrups t prvide additinal shear resistance beynd that prvided by the cncrete t prevent shear failure. 5

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