Example 04: Compressive Force in Concrete T-Beam

Problem
The following are the dimensions of a concrete T-beam section

Width of flange, bf = 600 mm
Thickness of flange, tf = 80 mm
Width of web, bw = 300 mm
Effective depth, d = 500 mm

 

The beam is reinforced with 3-32 mm diameter bars in tension and is carrying a moment of 100 kN·m. Find the total compressive force in the concrete. Use n = 9.
 

wsd-example-04-strength-of-t-beam.jpg

 

Design of Concrete Beam Reinforcement using WSD Method

Steps is for finding the required steel reinforcements of beam with known Mmax and other beam properties using Working Stress Design method.

Given the following, direct or indirect:

Width or breadth = b
Effective depth = d
Allowable stress for concrete = fc
Allowable stress for steel = fs
Modular ratio = n
Maximum moment carried by the beam = Mmax

 

wsd-doubly-reinforcement-design.jpg

Working Stress Analysis for Concrete Beams

Consider a relatively long simply supported beam shown below. Assume the load wo to be increasing progressively until the beam fails. The beam will go into the following three stages:

  1. Uncrack Concrete Stage – at this stage, the gross section of the concrete will resist the bending which means that the beam will behave like a solid beam made entirely of concrete.
  2. Crack Concrete Stage – Elastic Stress range
  3. Ultimate Stress Stage – Beam Failure
wsd-beam-analysis-crack-uncrack.jpg

 

Example 03: Moment capacity of a concrete beam

Problem
A reinforced concrete beam 300 mm wide has an effective depth of 600 mm. It is reinforced with 4-32 mm diameter bars for tension. f’c = 21 MPa and fy = 275 MPa. Find the moment capacity of the beam.
 

wsd-example-03-moment-capacity-beam.jpg

 

Example 02: Total compressive force in conrete

Problem
A rectangular reinforced concrete beam with width of 250 mm and effective depth of 500 mm is subjected to 150 kN·m bending moment. The beam is reinforced with 4 – 25 mm ø bars. Use alternate design method and modular ratio n = 9.

  1. What is the maximum stress of concrete?
  2. What is the maximum stress of steel?
  3. What is the total compressive force in concrete?

 

wsd-example-02-flexural-stresses-concrete-steel.jpg

 

Example 01: Required steel area of reinforced concrete

Problem
A rectangular concrete beam is reinforced in tension only. The width is 300 mm and the effective depth is 600 mm. The beam carries a moment of 80 kN·m which causes a stress of 5 MPa in the extreme compression fiber of concrete. Use n = 9.
1.   What is the distance of the neutral axis from the top of the beam?
2.   Calculate the required area for steel reinforcement.
3.   Find the stress developed in the steel.
 

wsd-example-01-unknown-steel-area.jpg

 

Properties of Wide-Flange Sections (W Shapes), SI Units

W = weight per linear length
A = cross-sectional area
d = overall depth
tw = thickness of web
tf = thickness of flange
bf = width of flange
I = moment of inertia
S = section modulus
r = radius of gyration
properties-of-section-w-flange.jpg

 

Example 02: Maximum concentrated load a box beam can carry

Problem
A beam is built up by nailing together 25 mm thick planks to form a 200 mm × 250 mm box section as shown. The nails are spaced 125 mm apart and each can carry a shearing force of up to 1.3 kN. The beam is simply supported for a span of 3.6 m and to carry a concentrated load P at the third point of the span. The allowable shearing stress of the section is 0.827 MPa.
 

ex-02-bolted-connection.jpg

 

  1. Determine the largest value of P that will not exceed the allowable shearing stress of the beam or the allowable shearing force of the nails.
  2. What is the maximum flexural stress of the beam for the load P computed above?

 

Example 03: Moment capacity of a timber beam reinforced with steel and aluminum plates

Problem
Steel and aluminum plates are used to reinforced an 80 mm by 150 mm timber beam. The three materials are fastened firmly as shown so that there will be no relative movement between them.
 

ex-03-bending-reinforced-timber-beam.jpg

 

Given the following material properties:

Allowable Bending Stress
Steel = 120 MPa
Aluminum = 80 MPa
Wood = 10 MPa
Modulus of Elasticity
Steel = 200 GPa
Aluminum = 70 GPa
Wood = 10 GPa

Find the safe resisting moment of the beam in kN·m.
 

Example 02: Required diameter of circular log for allowable shear stress

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.
 

diameter-of-log-for-shear-stress.jpg

 

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