OUTCOME 2 - TUTORIAL 4 REINFORCED CONCRETE BEAMS

Transcription

1 UNIT 6: Uni ode: QCF level: 5 Credi value: 15 STRENGTHS OF MATERIALS K/601/1409 OUTCOME - TUTORIAL 4 REINFORCED CONCRETE BEAMS Be able o deermine he behavioural haraerisis of loaded beams, olumns and srus Simply suppored beams: use of Maaulay s mehod o deermine he suppor reaions, slope and defleion due o bending in anilevers and simply suppored beams wih ombined onenraed and uniformly disribued loads Reinfored onree beams: heoreial assumpions; disribuion of sress due o bending Columns: sress due o asymmerial bending; middle hird rule for reangular seion olumns and walls; middle quarer rule for irular seion olumns Srus: end fixings; effeive lengh; leas radius of gyraion of seion; slenderness raio; Euler and Rankine-Gordon formulae for deerminaion of riial load You should judge your progress by ompleing he self assessmen exerises. CONTENTS 1. INTRODUCTION. ELASTIC BENDING. CONCRETE BEAMS 4. SECOND MOMENT OF AREA 5. PLASTIC BENDING I is assumed ha sudens doing his uorial are fully onversan wih bending sresses in beams. D.J.Dunn Maerial supplied from 1

2 1. INTRODUCTION In order o undersand he heory of reinfored onree beams you would need o sudy a very wide range of maerial well beyond he sope of one par of one ouome of one module. For his reason he maerial offered here is grealy simplified.. ELASTIC BENDING Consider a simple reangular beam a a poin where he bending momen is M. The neural axis passes hrough he enre of area. The srain is direly proporional o disane y. Sine σ = E ε he sress is also direly proporional o disane y. Suffix refers o ompression and o ension. From he bending formula we know ha he sress a he ouer edge is σ = M y/i = M/Z y = D/ a he ouer edges. I is he seond momen of area abou he enroid. Z = I/y = I/D is he elasi modulus. (This may be looked up in sandard ables for ommerial beams of various seions). M = σ Z where σ is he aual value of sress a he ouer edge so long as i is wihin he elasi limi.. CONCRETE BEAMS A beam made from pure onree would fail very easily beause onree anno wihsand ension so a rak would open up on he ensile edge and spread hrough he seion. To sop his happening, seel rods are embedded in he onree near o he ensile edge. These are onvolued o help preven hem slipping in he onree. We assume ha he rods are so firmly embedded in he onree ha hey beome srained by he same amoun as he onree and alhough he onree would break, he rods will bridge he gap and preven he rak from opening The onree under ension does no produe any fore. The enire ensile fore is arried by he seel rods. D.J.Dunn Maerial supplied from

3 The diagram shows how he sress and srain are disribued in he onree and seel reinforemen. A s is he seion area of he rods. The problem now is ha he neural axis moves away from he enroid and o find y we need o sudy how o find he seond momen of area when we have wo maerials. 4. SECOND MOMENT OF AREA The onree under ompression is he shaded area and he ompression fore aing on his area mus be equal o he ensile fore in he seel rods. Beause of his, he neural axis is no longer a he middle and we need a way of deermining y. For he onree seion only, I = B(D - y ) / abou he neural Axis. Assuming he seel is onfined o one level and ha y is measured from his posiion, I s = A s y This is seel and no onree and we need o find he equivalen seond momen of area for he equivalen area of onree. Consider a seion of seel on whih he sress is σ. The Fore is F = A s σ and sine σ = ε E The sress is σ s = A s ε E s for elasi maerials. Suppose he seel was replaed wih an area of onree of equal srengh and he same srain. F = A ε E = A s ε E s A = A s ε E s / ε E = A s E s /E = n A s n = E s /E This is he equivalen area of onree and he equivalen seond momen of area is I s = n A s y The seond momen of area for he seion in equivalen onree erms is y B D INA nasy Now onsider he fore balane again. The ensile fore in he rods is F = A s σ = A s E ε The ompressive fore in he onree is (D - y ) B σ / = (D - y ) B E ε / The ensile and ompressive fores mus be equal. A s E ε = (D - y ) B E ε / ε ε As E nas...(1) B D y E BD y D.J.Dunn Maerial supplied from

4 From he wo similar riangles in he srain disribuion we have ε ε ε D y hene...() D y y ε y D y nas Equae (1) and () y BD y This may be rearranged ino a quadrai equaion. nya s nya s D y D y Dy 0 B B As y yd n D 0 B Now use he quadrai formula o solve b y y y ρ b 4a a As D n B y 4 4D ρ Dρ D a 1 As 4 D n B D As b D n B 4D As le D n B D ρ WORKED EXAMPLE No.1 A reangular reinfored onree beam has he following daa. D = 155 mm B = 10 mm n = 8 A s = 40 mm Find he posiion of he neural axis and he seond momen of area abou i. SOLUTION Suffix refers o onree and s o seel. As 40 D n B 10 y y 98.8 mm from heseel rods. I I NA NA y B D nasy x 40 x x x x 10 mm D.J.Dunn Maerial supplied from 4

5 WORKED EXAMPLE No. For he same beam as example 1, he maximum srain in he onree mus no exeed Deermine he maximum ompressive sress in he onree, he srain in he seel and he sress in he seel. Show ha he ensile and ompressive fores are equal. Assume he srain disribuion is linear. Take E = 5 GPa SOLUTION E s = n E = 8 x 5 = 00 GPa 9 σ E ε 5 x 10 x MPa ε σ s yε 98.8 x 10 x D y 155 x x 10 E ε s s x 10 x MPa F = B(D - y)σ / = {1 ( )} x 10-6 x 75 x 10 6 / = 5.9 kn (ompressive) F s =A s σ s /= 40 x 10-6 x 1054 x 10 6 = 5.9 kn (Tensile) SELF ASSESSMENT EXERCISE No.1 A reangular reinfored onree beam has he following daa. D = 94 mm B = 05 mm n = 8 A s = 1550 mm Find he posiion of he neural axis and he seond momen of area abou i. The maximum srain in he onree mus no exeed Deermine he maximum ompressive sress in he onree, he srain in he seel and he sress in he seel. Assume he srain disribuion is linear. Take E = 5 GPa Answers y = 51 mm from he seel rods. I NA = x 10 9 mm 4 σ = 1.5 MPa ε s = σ s = MPa D.J.Dunn Maerial supplied from 5

6 The work overed so far is no a very realisi approah o reinfored beams and we should onsider he heory of plasi failure and rue sress disribuion in order o make a realisi sab a solving beam problems. Mos published ex on reinfored onree beams sae ha onree fails in ompression when he srain is Furher sudies reveal ha he seel will usually have passed is ensile yield poin. The ype of seel used is assumed o have a plasi range and so he sress in he seel will be he yield sress. Typial seel has a yield sress of around 415 MPa and an elasi modulus E of 00 GPa. The srain a he yield poin is hene around I is imporan o hek ha he seel has yielded if he yield sress value is used. 5. PLASTIC BENDING If he maerial reahes is yield sress a some value of y less han D/, he sress is no longer proporional o srain. To simplify maers le s assume ha he sress says onsan a he yield value σ y for all srains beyond he yield poin. The diagram ompares pure plasi yielding wih ha of ypial seel. The sress and srain disribuion is now as shown. Noe ha he srain disribuion is unhanged sine srain is direly proporional o he disane from he neural axis. I may be of ineres o noe ha for seel beams, yielding firs ours a he ouer edge and he bending momen is M = σ y Z p where Z p is he plasi modulus. This may also be found in ables for ommerial seions bu his is no muh use for reinfored beams. The relaionship beween sress and srain is no linear for he onree. Alhough he srain disribuion remains he same as before, he sress disribuion is more like his. D.J.Dunn Maerial supplied from 6

7 In order o work ou he ompressive fore in he onree, he sress disribuion is simplified o a box as shown. This is known as he Whiney box. The ompressive sress is aken as a reangle of heigh a and lengh 0.85 σ The dimension a is usually given as a = β 1 (D - y ) The ompressive fore F as a he middle of he box a a disane (D - y - a/) from he neural axis. I is normal o ake β 1 = 0.85 for σ 0 MPa and 0.65 for larger sress values. The ompressive fore is now given by he expression F = 0.85B a σ Equaing ensile and ompressive fore we have A s σ = 0.85B a σ Asσ a 0.85Bσ The bending momen is M = F (D - a/) = F (D - a/) M = F (D - a/) = A s σ s (D - a/) Subsiue for a and Noe he following poins. M A σ s Asσ D x 0.85Bσ Asσ 0.59Asσ D Bσ The ensile srengh in he onree is always ignored. The seel rods may have yielded in whih ase he σ = σ y Conree is assumed o have failed in ompression when he srain reahes The srain disribuion is assumed o be linear. There are limiaions o he formulae developed here and more advaned ex should be sudied before applying hem o real siuaions. D.J.Dunn Maerial supplied from 7

8 WORKED EXAMPLE No. A reangular onree beam has he following dimensions. B = 05 mm D = 94 mm (The disane from he op o he reinforemen). Yield sress for onree = 7.6 MPa Yield sress for seel = 415 MPa The seel rods have a ross seional area of 1550 mm Assume ha boh he seel and onree have yielded. Find he neural axis. Find he momen apaiy. Given ha he srain in he onree is 0.00 a yield, verify ha he seel has also yielded. Take E s = 00 GPa SOLUTION Asσ a and if boh have yielded we pu in he yield values. 0.85Bσ 1550 x 10 x 415 x x 05 x 10 x 7.6 x 10 Now find he neural axis. a m or 89.9 mm β 1 (D - y ) = a and sine σ 0 MPa β 1 = 0.85 a = 89.9 = 0.85(94 - y ) y = 88. mm from he boom edge. Based on he seel he momen apaiy is: M = A s σ s (D - a/) = 1550 x 10-6 x 415 x 10 6 ( /) x 10 - = 4 knm The srain in he onree is found from he linear relaionship. Ε s / ε = y /(D- y ) = 88. /(94 88.) =.7 ε s = 0.00 x.7 = The srain in he seel a yield is ε s = σ y /E = 415 x 10 6 /00 x 10 9 = The seel has exeeded he yield poin. D.J.Dunn Maerial supplied from 8

9 SELF ASSESSMENT EXERCISE No. 1. A reangular reinfored onree beam has he following daa. D = 600 mm B = 400 mm Yield sress for onree = 0 MPa Yield sress for seel = 40 MPa The seel rods have a ross seional area of 000 mm Assume ha boh he seel and onree have yielded. Find he neural axis. Find he momen apaiy. Given ha he srain in he onree is 0.00 a yield, verify ha he seel has also yielded. Take E s = 00 Gpa (Answers 455 mm from he reinforemen, 678 knm). A reangular reinfored onree beam has he following daa. D = 05 mm B = 5.4 mm Yield sress for onree = 8 MPa Yield sress for seel = 414 MPa The seel rods have a ross seional area of 196 mm Assume ha boh he seel and onree have yielded. Find he neural axis. Find he momen apaiy. Given ha he srain in he onree is 0.00 a yield, alulae he aual srain and he yield poin srain and verify ha he seel has also yielded. Take E s = 00 GPa (Answers 40 mm from he reinforemen, 8. knm, and 0.001) D.J.Dunn Maerial supplied from 9