Video from my YouTube channel

Subscribe to My You tube channel

Example 3: design of rectangular beam

By -
design rectangular sections for the beams, loads, and ρ values shown. Beam weights are not included
in the loads shown.  Assume concrete weighs 150 lb/ft3 , fy= 60,000 psi, and fc'= 4000 psi, unless given otherwise.

Figure 1








first, we will calculate the ultimate load, we have point live load and distributed dead load, therefore will take each load separately

ultimate live load=1.6*PL=1.6*20=32k

assuming the weight of beam=0.50k/ft

dead load=2+0.5=2.5 k/ft

ultimate dead load=1.2*2.5=3.00k/ft

Mu from moment diagram in figure 1=657.5k-ft=7,890,000lb-in

                                                                               Figure 1



Mu=Ф*Mn=Ф*Asfy(d-a/2)    

Mu/(b*d^2)=Ф*ρ*fy(1-fy*ρ/1.7*fc')    

ρ=0.18*4,000/60,000=0.012

bd^2=Mu/(Ф*ρ*fy(1-fy*ρ/1.7*fc')   assuming Ф=0.90

Mu=7,890,000lb-in

bd^2=7,890,000/(0.9*.012*60,000*(1-60,000*0.012/1.7*4000))=13,617.88

taking bd=    16x29.5
          

using 16*33 as beam dimension (bxh)

checking beam weight=16*33*150/144=0.55k/ft    not ok

then calculating ultimate dead load again

dead load=2+0.55=2.55 k/ft

ultimate dead load=1.2*2.55=3.06k/ft

Mu=664.25 k-ft=7,971,000lb-in

bd^2=7,971,000/(0.9*.012*60,000*(1-60,000*0.012/1.7*4000))=13,757.61

bd=    16x29.5 is ok 

As=ρ*b*d=0.012*16*29.5=5.664in2

 using 4#11=4*1.56=6.24 in2   ok


checking for Ф=0.9 or no

(εt+0.003)/(d)=0.003/c

C=a/β1

β1=0.85 for fc'=4000 psi or less

 a=Asfy/(0.85*fc'*b)

a=6.24*60,000/(0.85*4000*16)=6.88in

C=a/β1=6.88/0.85=8.1in

εt=0.003*(29.5)/8.1-0.003=0.0079>0.005 so section is tension control Ф=0.9  is ok

checking for ρmin

As,min=(3√fc'/fy)*bw*d and not less than 200bw*d/fy

As,min=3*√4000*16*29.5/60,000  and not less than 200*29.5*16/60,000

As,min=1.49 in2 and not less than 1.57in2 ok, our As is larger than the minimum 

checking Ф*Mn>Mu

Ф*Mn=0.9*6.24*60,000(d-a/2)


Ф*Mn=0.9*6.24*60,000(29.5-6.88/2)=8,781,177.6lb-in> Mu ok 





Comments

Post a Comment

Videos from My Youtube channel

Popular posts from this blog

Pile cap

By -
Image
Pile cap used to transfer the loads from superstructure to the piling. The pile cap is thick concrete mat rests on piles. It is part of the foundation and used to distribute the loads over the piles. Piles used when the soil bearing is not enough to carry loads of the structure. Figure 1  construction management: concrete construction bridge construction:How to become a bridge engineer Pile cap construction procedure Excavation around piles: After 7 days of the casting of the pile the concrete will gain approximately, 70% of it is compressive strength. The soil around piles will be removed to the required levels and dimension that allows construction of the pile cap. A step shall be made if the excavation is more than 1.5 m to prevent soil collapse. Figure 2 Pile head cutting: Concrete will be removed up to cut off level. If concrete under the cut off level is unsound. It shall be removed, and the pile will be repaired...
Read more

Bridge Diaphragm

By -
Image
The main function of diaphragms is to provide a stiffening effect to deck slab in case bridge webs are not situated directly on top of bearings. Therefore, diaphragms may not be necessary in case bridge bearings are placed directly under the webs because loads in bridge decks can be directly transferred to the bearings.On the other hand, diaphragms also help to improve the load-sharing characteristics of bridges. Diaphragms also contribute to the provision of torsional restraint to the bridge deck. Diaphragm will be used also to resist stresses that occurs during the post-tensioning process. Post-tensioning hardware are casted inside the diaphragm. Figure 1 construction management: concrete construction bridge construction:How to become a bridge engineer Types of diaphragm: Diaphragm at the span end: this type of diaphragms will be located between two bridge spans or at the bridge abutments. Expansion joints will be used to connect bridge spans. This ...
Read more

stress-strain diagram for aluminium alloy and rubber

By -
Image
figure no:1 showing the stress-strain diagram for aluminum alloy. we can notice from the stress-stress diagram that aluminum has a considerable ductility. despite the absence of explicit yield point as in structural steel . the initial portion of the stress-strain curve is linear with a recognizable proportional limit. the proportional limit for aluminum alloy ranges from 70 to 410 Mpa (10 to 60 Ksi). aluminum alloy undergoes large strain before failure. ultimate stress ranges from 140 to 550 Mpa (20 to 80 Ksi). construction management: concrete construction bridge construction:How to become a bridge engineer Figure-1 Yield stress for aluminum alloy can be determined by drawing offset line parrel to the linear portion of the stress-strain curve. the straight line offset by a standard strain such as 0.002. the intersection point with the stress-strain curve is the point of yield stress. figure no:2 illustrates the concept of offset method. yield stress acquire...
Read more

Chamfers for concrete structures

By -
Image
In general, the edges of concrete members are chamfered. Usually during the process of installing forms for a structural member such as pile cap, a chamfer a PVC chamfer as shown in figure no:1 installed at the edges of the structural members. These chamfers will form smooth slanted edges, as shown in figure no:2. The size of chamfers can be 20x20mm or more as shown on the contract drawing. The PVC chamfer fixed to forms by nails. We should ensure that the PVC chamfer fixed in their correct locations, and we should ensure that the PVC chamfer is straight and not moving up or down. Checking the alignment and location of PVC chamfer is important, so the chamfered concrete edge will have a good appearance after the form’s removal.  Figure 1 Figure 2 Chamfering the concrete edges done for many reasons such as aesthetics, some suggesting that the chamfering of concrete edges done for safety to reduce the damage of objects due to impact with sharp concrete edges. Actually, ...
Read more

Flakiness Index and Elongation Index of Coarse Aggregates

By -
Image
Flaky and elongated aggregates can adversely affect concrete. Flaky and elongated aggregates are difficult to compact. Therefore using flaky and elongated aggregates will reduce concrete workability, which required more amount of water to produce a workable mix. Increasing water content will reduce concrete strength and durability. Moreover, these aggregates are not robust, which leads to lowering concrete strength. Figure 1 construction management: concrete construction bridge construction:How to become a bridge engineer The Flakiness Index of aggregates is the percentage by weight of particles whose least dimension (thickness) is less than three- fifths (0.6times) of their mean dimension. This test does not apply to sizes smaller than 6.3mm. The Elongation index of an aggregate is the percentage by weight of particles whose greatest dimension (length) is greater than nine-fifths (1.8times) their mean dimension. Flakiness and elongation tests ar...
Read more

Example 1:Flexural stress calculation using transformed area method for rectangular beam with double layer of reinforcing bars

By -
Image
assume the section for beam shown in figure 1 cracked calculate the flexural stresses for the given load or moment using transformed area method. construction management: concrete construction bridge construction:How to become a bridge engineer Figure 1 first, we will calculate the transformed area of steel. the concrete below the neutral axis will not contribute to resisting of tension. figure 2 shows the section after transformation  As=6*1=6.0in2 Transformed area for first layer=As*n=2*8=16in2 Transformed area for second layer=As*n=4*8=32in2 figure 2 Neutral axis can be calculated by equalling moment of area from both parts as shown below  16*(18-x)+32*(21-x)=(x/2)*(14*x) 960-48x=7x^2 7x^2+48x-960=0 x=8.77 in moment of inertia for transformed area= =(14*8.77^3)/12+8.77*14*4.385^2+16*9.23^2+32*12.23^2=9297.20in4 =0.44835ft4 flexural stress for compression zone. y at extreme compression equal 8.77in=0.73ft fc...
Read more

why do we use cranked rebar at rebar overlapping?

By -
Image
 Cranked bars are produced by slightly bending the reinforcement bar before the overlapping zone then straightening it at the overlapping zone, as shown in figure no:1. The main purpose of this practice is to reduce the congestion of reinforcement at the overlapping zone for structure members with a high amount of reinforcement. The use of crank bars will give flexibility in fixing and arranging reinforcement to reduce the congestion, which helps create a larger space between reinforcement rebars. The spacing between reinforcement rebars should be sufficient to allow smooth concrete placement. Small spacing of reinforcement rebar will work as a screen that will enable the concrete paste to pass through it and stop aggregate, which results in concrete segregation. Therefore, cranked rebars can be used to slightly reduce the congestion of reinforcement rebars and increase the spacing between rebars. Figure no:1
Read more

Repairing of concrete bulging

By -
Image
Improper designing and installation of formwork for a concrete member can result in concrete Bulging. During the placing and compacting of concrete, forms may deflect largely in the weak areas. The deflection of forms will cause concrete to expand at this location, and this known as concrete Bulging. Concrete Bulging will spoil the appearance of a concrete member. Therefore we should always ensure that the forms are appropriately installed, and it is sufficiently supported to resist the concrete during placing and compacting. Bulging should be repaired in case we couldn't avoid it. The solution of concrete Bulging, offset, or surface irregularities are surface grinding. However, excessive surface grinding can weaken the concrete.  Figure 1 If the bulging or surface irregularities are small, grinding of concrete surface will be sufficient to remove surface irregularities. If the Bulging of concrete is significant, then the concrete should be chipped, and the surface...
Read more

prestressing equipment

By -
Image
Jacks are essential tools in prestressing operations and are used to apply tension to strands within tendons. There are two main types of jacks: A. Mono-Strand Jack A mono-strand jack is primarily used in building construction, where tendons typically consist of a small number of strands (usually 3 to 4). This type of jack is designed to stress each strand individually, ensuring precise control over the tensioning process. Since buildings generally have shorter tendons compared to bridges, stressing strands one by one is a practical and efficient approach in this context. construction management: concrete construction bridge construction:How to become a bridge engineer B. Multi strands jack:  In bridge construction, where tendons contain a significantly larger number of strands, using a mono-strand jack would be impractical. Stressing strands one by one in such cases would not only be time-consuming but also increase the risk of strand entrapment. Ther...
Read more

Whitney Rectangular Stress Distribution(derivation of beam expressions)

By -
Image
experimental tests confirm that the strain varies in proportion to a distance from the neutral axis in compression and tension zone within and near the ultimate stress. compression stress will vary linearly as long the maximum stress is less than 0.5fc'. once the stress exceeds 0.5fc' the distribution of compression stress will change from linear distribution to the shape shown in figure 1. compressive stress will vary from zero at the neutral axis to a maximum value at or near the extreme fiber. neutral axis location and stress variation will differ from a beam to another. this variation depends on several factors such as magnitude and history of post loading, shrinkage, creep and speed of loading. construction management: concrete construction bridge construction:How to become a bridge engineer Figure 1 for simplification whiteny replaced the curved stress block with a rectangular stress block as shown in the figure 2. compressive stress assumed to ...
Read more