Question

A 1200 mm deep by 750 mm wide post-tensioned simply supported beam is shown below. The beam spans 12.0 m and is subject to a superimposed dead load of 50 kN/m and a live load of 35 kN/m. Both the superimposed dead load and live load are applied after transfer (after stressing has taken place). The tendon is located at the mid-height of the beam at each end, and its centreline sits 50 mm from the base at midspan. The concrete strength at transfer is 22 MPa, and at maturity is 40 MPa. Assume E_c = 32800 MPa, γ_c = 24 kN/m³ and ignore any prestress losses.

a)            If P_i = 2700 kN, assess the adequacy of the beam at transfer. In addition, assuming P_i cannot be changed, briefly describe three methods that could be used to ensure the transfer stresses are within code limits.

Answer 1

Solution:- the values given in the question are as follows:

depth of beam(d)=1200 mm

width of beam(b)=750 mm

length of beam(L)=12 m

dead load(D.L)=50 kN/m

live load(L.L)=35 kN/m

eccentricity of tendon at centre(e)=50 mm

stress transfer in concrete(I)=22 MPa

maturity stress in concrete()=40 MPa

young's modulus of elasticity of concrete(Ec)=32800 MPa

unit weight of concrete()=24 kN/m^3

imposed post-tensioned force(Pi)=2700 kN

(a) , Stress concept method:

direct stress due to post-tensioned force=P/A

Pi=P=2700 kN or Pi=2700*1000 N

A=1200*750=900000 mm^2

direct stress due to post-tensioned force=2700000/900000=3 N/mm^2

stress due to post-tensioned force with eccentricity=P*e/Z=0

Pi passing through the center line of beam so the stress due to eccentricity is zero(0)

stress due to external bending moment=M/Z

M=W*L^2/8=

M=85*12000^2/8=1.53*10^9

Z=(b*d^3/12)/(d/2)=b*d^2/6

Z=750*1200^2/6=18*10^7 mm^3

stress due to external bending moment=(1.53*10^9)/(18*10^7)=8.5 N/mm^2

total stress at top=3+8.5=11.5 N/mm^2

total stress at bottom=3-8.5=-5.5 N/mm^2

transfer stress in concrete is 20 N/mm^2 so the stress 11.5 N/mm^2 is with in the limit.

(b) , Load balancing concept:

let resisting force developed in tendon due to post-tensioned force applied on tendon is q , shown in figure.

bending moment at every section of tendon is zero because the post-tension force is balanced by , resisting force(q) developed in tendon.

bending moment at centre of beam is also zero

P*e=q*L/2*L/4

q=8P*e/L^2

q=(8*2700*50*10^-3)/(12^2)

q=7.5 kN/m

net bending moment at center={(W-q)*L^2}/8={(85-7.5)*12^2}/8

net bending moment at center(net BM)=1395 kN-m

net bending moment at center(net BM)=1395*10^6 N-mm

extreme stress=P/Anet BM/Z

extreme stress=3(1395*10^6)/(18*10^7)

extreme stress37.75

stress at top fiber of beam=3+7.75=10.75 N/mm^2

stress at bottom fiber of beam=3-7.75=-4.75 N/mm^2

transfer stress in concrete is 20 N/mm^2 so the stress 10.75 N/mm^2 is with in the limit.

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