Problem 4.5. Deflection of a RC beam

The cross section of a simply supported beam is given in the previous example. Length of the beam is L = 4 m. Determine the deflection at midspan. The beam is subjected to

a) a uniformly distributed load, p = 16 kN/m,
b) a concentrated load at the midspan, P = 32 kN.

(When M  <  Mcr use the uncracked cross section, while for M  ≥ Mcr use the cracked cross section.)

Solve Problem

Solve

Problem a)

Midspan deflection, v [mm]=

Problem b)

Midspan deflection, v [mm]=

Do you need help?

Steps

Step by step

Problem a)

Step 1. Calculate the maximum (midspan) moment from the uniformly distributed load, p. Check whether the cross section is cracked.

Check moment

M=pL28=16×428=32 kNm>Mcr=15.41 kNm

The cross section cracks. Cracking moment is determined in the previous problem (Problem 4.4). The cracked cross sectional properties must be used in the deflection analysis, which are also calculated in the previous problem.

Step 2. Calculate the midspan deflection.

Show deflection

See Table 3.5.
v=5384pL4EcIe,II=538416×4000418.3×103×1.604×109=1.82 mm

Problem b)

Step 1. Calculate the maximum (midspan) moment from the concentrated load, P. Check whether the cross section is cracked.

Check moment

M=pL4=32×44=32 kNm>Mcr=15.41 kNm

The cross section cracks. Cracking moment is determined in the previous problem (Problem 4.4). The cracked cross sectional properties must be used in the deflection analysis, which are also calculated in the previous problem.

Step 2. Calculate the midspan deflection.

Show deflection

See Table 3.5.

v=148PL3EcIe,II=54832×4000318.3×103×1.604×109=1.45 mm

Results

Worked out solution

Problem a)

The maximum (midspan) moment from the uniformly distributed load, p is

M=pL28=16×428=32 kNm>Mcr=15.41 kNm

The cross section cracks. Cracking moment is determined in the previous problem (Problem 4.4). The cracked cross sectional properties must be used in the deflection analysis, which are also calculated in the previous problem.

The midspan deflection is 

See Table 3.5.
v=5384pL4EcIe,II=538416×4000418.3×103×1.604×109=1.82 mm

Problem b)

The maximum (midspan) moment from the concentrated load, P is

M=pL4=32×44=32 kNm>Mcr=15.41 kNm

The cross section cracks. Cracking moment is determined in the previous problem (Problem 4.4). The cracked cross sectional properties must be used in the deflection analysis, which are also calculated in the previous problem.

The midspan deflection is

See Table 3.5.

v=148PL3EcIe,II=54832×4000318.3×103×1.604×109=1.45 mm