Example 01: Total Compression Force at the Section of Concrete Beam

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-01-flexural-stresses-concrete-steel.jpg

 

Example 01: Required Steel Area of Reinforced Concrete Beam

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

 

Design of Steel Reinforcement of Concrete Beams by 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-reinforced-beam.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

 

Consistency of Soil (Atterberg Limits)

Consistency is the term used to describe the ability of the soil to resist rupture and deformation. It is commonly describe as soft, stiff or firm, and hard.
 

Water content greatly affects the engineering behavior of fine-grained soils. In the order of increasing moisture content (see Figure 2 below), a dry soil will exist into four distinct states: from solid state, to semisolid state, to plastic state, and to liquid state. The water contents at the boundary of these states are known as Atterberg limits. Between the solid and semisolid states is shrinkage limit, between semisolid and plastic states is plastic limit, and between plastic and liquid states is liquid limit.
 

002-atterberg-limits.gif

 

Atterberg limits, then, are water contents at critical stages of soil behavior. They, together with natural water content, are essential descriptions of fine-grained soils.
 

Physical Properties of Soil

Phase Diagram of Soil
Soil is composed of solids, liquids, and gases. Liquids and gases are mostly water and air, respectively. These two (water and air) are called voids which occupy between soil particles. The figure shown below is an idealized soil drawn into phases of solids, water, and air.
 

001-phase-diagram-of-soil.gif

 

Weight-Volume Relationship from the Phase Diagram of Soil
total volume = volume of soilds + volume of voids
$V = V_s + V_v$

volume of voids = volume of water + volume of air
$V_v = V_w + V_a$

total weight = weight of solids + weight of water
$W = W_s + W_w$
 

Three Reservoirs Connected by Pipes at a Common Junction

Situation
Three reservoirs A, B, and C are connected respectively with pipes 1, 2, and 3 joining at a common junction P. Reservoir A is at elevation 80 m, reservoir B at elevation 70 m and reservoir C is at elevation 60 m. The properties of each pipe are as follows:

Pipe 1:   L = 5000 m, D = 300 mm
Pipe 2:   L = 4000 m, D = 250 mm
Pipe 3:   L = 3500 m

The flow from reservoir A to junction P is 0.045 m3/s and f for all pipes is 0.018.
 

011-three-reservoir-problems.jpg

 

  1. Find the elevation of the energy grade line at P in m.
    A.   75.512
    B.   73.805
    C.   72.021
    D.   74.173
  2. Determine the flow on pipe 2 in m3/s.
    A.   0.025
    B.   0.031
    C.   0.029
    D.   0.036
  3. Compute the diameter appropriate for pipe 3 in mm.
    A.   175
    B.   170
    C.   178
    D.   172

Problem 20 - Bernoulli's Energy Theorem

Problem 20
The 600-mm pipe shown in Figure 4-11 conducts water from reservoir A to a pressure turbine, which discharges through another 600-mm pipe into tailrace B. The loss of head from A to 1 is 5 times the velocity head in the pipe and the loss of head from 2 to B is 0.2 times the velocity head in the pipe. If the discharge is 700 L/s, what power is being given up by the water to the turbine and what are the pressure heads at 1 and 2?
 

04-014-flow-with-turbine.gif

 

Problem 19 - Bernoulli's Energy Theorem

Problem 19
A pump draws water from reservoir A and lifts it to reservoir B as shown in Figure 4-10. The loss of head from A to 1 is 3 times the velocity head in the 150-mm pipe and the loss of head from 2 to B is 20 times the velocity head in the 100-mm pipe. Compute the horsepower output of the pump and the pressure heads at 1 and 2 when the discharge is: (a) 12 L/s; (b) 36 L/s.
 

04-013-flow-with-pump.gif

 

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