circular beam

Situation
A concrete beam with cross section in Figure CO4-2B is simply supported over a span of 4 m. The cracking moment of the beam is 75 kN·m.
 

figure-co4-2b.jpg

 

1.   Find the maximum uniform load that the beam can carry without causing the concrete to crack, in kN/m.

A.   35.2 C.   33.3
B.   37.5 D.   41.8

2.   Find the modulus of rapture of the concrete used in the beam.

A.   4.12 MPa C.   3.77 MPa
B.   3.25 MPa D.   3.54 MPa

3.   If the hallow portion is replaced with a square section of side 300 mm, what is the cracking moment of the new section in kN·m?

A.   71.51 C.   78.69
B.   76.58 D.   81.11

 

Example 02: Required Diameter of Circular Log Used for Footbridge Based on Shear Alone

Problem
A wooden log is to be used as a footbridge to span 3-m gap. The log is required to support a concentrated load of 30 kN at midspan. If the allowable stress in shear is 0.7 MPa, what is the diameter of the log that would be needed. Assume the log is very nearly circular and the bending stresses are adequately met. Neglect the weight of the log.
 

beam-001-circular-log-shear-stress.gif

 

Bending Stress and Shearing Stress in Timber Beam

Bending Stress
$f_b = \dfrac{M}{S} = \dfrac{Mc}{I}$
 

Horizontal Shear Stress
$f_v = \dfrac{VQ}{Ib}$

 bending-member-corss-section-any-shape.gif
For Rectangular Sections
$f_b = \dfrac{6M}{bd^2}$
 

$f_v = \dfrac{3V}{2bd}$

Solution to Problem 568 | Horizontal Shearing Stress

Problem 568
Show that the shearing stress developed at the neutral axis of a beam with circular cross section is τ = (4/3)(V / π r2). Assume that the shearing stress is uniformly distributed across the neutral axis.
 

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