Question

A concrete beam prestressed with a parabolic tendon is shown in the sketch. The eccentricity of the tendon is measured from the centroid of the cross-section. The applied prestressing force at service is 1620 kN. The uniformly distributed load of 45 kN/m includes the self-weight.




The stress (in N/mm${}^2$) in the bottom fibre at mid-span is

• A. tensile 2.90
• B. compressive 2.90
• C. tensile 4.32
• D. compressive 4.32

Answer 1

The correct option is B compressive 2.90


Moment at the centre due to UDL,

$M_u=\frac{w{l}^2}{8}=\frac{45\times (7.3{)}^2}{8}$

= 299.76 kNm
$=299.76\times {10}^{6}Nmm$

Cross section area,
$A=500\times 750m{m}^2$

Moment of inertia about neutral axis,

$I=\frac{b{d}^3}{12}=\frac{500\times {750}^3}{12}$

$=1.757\times {10}^{10}m{m}^4$

Prestressing force, P = 1620 kN
$=1620\times {10}^3$ N
Eccentricity at mid-span ,
e = 145 mm
Distance of bottom fibre from neutral axis,
$y=\frac{750}{2}=375mm$

Stress in the bottom fibre at mid span

$\sigma =\frac{P}{A}+\frac{(P.e).y}{I}-\frac{M_{u}y}{I}$

$=\frac{1620\times {10}^3}{500\times 750}+\frac{(1620\times {10}^3\times 145).375}{1.757\times {10}^{10}}-\frac{299.76\times {10}^6\times 375}{1.757\times {10}^{10}}$

= 4.32 + 5.014 - 6.398

= 2.9 compressive