Question

An I shaped prestressed concrete beam spanning 30 m has an effective prestress (just after transfer) of 3000 kN at an eccentricity of 75 mm at midspan cross-section. Calculate the stresses at midspan in the extreme fibres. Given the following data:
Area of cross-section $=0.42m^2$
Moment of inertia $=0.1433m^4$
Height of the beam = 1600 mm
C.G. of the section from the bottom = 890 mm
$Z_{top}=0.202m^3$
$Z_{bottom}=0.161m^3$
Density of concrete $=25kN/m^3$

Answer 1

Dead load of the beam $w_d=A\times {\gamma }_c$
$=0.42\times 25$
$=10.5$ kN/m
B.M at midspan due to dead load ($M_d$)
$=\frac{w_d\times l^2}{8}=\frac{10.5\times {30}^2}{8}$
= 1181.25 kNm
Effective prestress (P) = 3000 kN
Eccentricity (e) = 75 mm

Stress at top
$=\frac{P}{A}-\frac{Pe}{Z_t}+\frac{M_d}{Z_t}$

$=\frac{3000\times {10}^3}{0.42\times {10}^6}-\frac{3000\times {10}^3\times 75}{0.202\times {10}^9}+\frac{1181.25\times {10}^6}{0.202\times {10}^9}$

$=11.87N/m{m}^2$

Stress at bottom
$=\frac{P}{A}+\frac{Pe}{Z_b}-\frac{M_d}{Z_b}$
$=\frac{3000\times {10}^3}{0.42\times {10}^6}+\frac{3000\times {10}^3\times 75}{0.161\times {10}^9}-\frac{1181.25\times {10}^6}{0.161\times {10}^9}$
$=1.203N/m{m}^2$