Video from my YouTube channel

Subscribe to My You tube channel

Conjugate beam Problem no:5

By -

Example 5: determine the displacement at C and the slope at point B for the beam shown in figure no: 1 considering EI a constant?

Figure 1

 

The support at A for this beam will allow the rotation but no movement, so it is considered a pin, and it will stay the same in the conjugate beam. While the support at B will allow the movement in horizontal direction and rotation, it will be similar to the roller, and it will stay the same in the conjugate beam. So the real beam will be as in figure no:2. And the conjugate beam will be as shown in figure no:3.

Figure 2

Figure 3





To determine the displacement at point C and the slope at B, we need to determine the moment at point C and the shear at point B in the conjugate beam. The conjugate beam will be loaded with M/EI of the real beam, so we need to draw M/EI for the real beam. To draw the M/EI diagram, we need to determine the reaction at point A and point B.

From the symmetry we can know that the reaction FA=FB=P. so the shear moment diagram will be as shown in the figure no:4.

Figure 4



So the conjugate beam will be loaded as shown in the figure no:5. 
Figure 5


From the symmetry the reaction FA=FB. So



 







To determine the displacement at point C, we need to determine the moment, we will take the section A-C. so using the equilibrium equations





















Figure 6

Figure 7




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

Rebar Splicing and Overlapping — ACI 318-19 Requirements Explained

By -
Image
Reinforcement splicing or overlapping is a fundamental requirement in reinforced concrete structures. Because reinforcement bars (rebars) are manufactured in limited standard lengths—typically 12 m —it becomes necessary to splice or overlap them to achieve the required continuity in structural members. The main purpose of a splice is to ensure smooth stress transfer between adjacent bars through the surrounding concrete. The overlap length, commonly known as the splice length , must be long enough to safely transfer the tensile stresses from steel to concrete without causing bond failure. It is recommended to avoid splicing in high-stress zones such as areas of maximum moment, and to stagger the splices to prevent congestion and reduce stress concentration. The ACI 318-19 providing the following recommendatin and requirement for the lap splicing: 1. Splice Limitation by Bar Size Lap splicing is not permitted for reinforcement bars larger than 36 mm in diameter. 2. Non-Contact L...
Read more

Analysis of T Beams

By -
Image
Analysis of T beam is similar to rectangular beam. same equations will be used if 'a' is less than the depth of flange hf. "a" is the depth of compression block. if "a" is larger than hf. compression stress will be distribuated over flange and web. flange and web has different width. therefore the compressive force C=0.85*fc'*hf*bf+0.85*fc'*(a-hf)*bw as shown in figure 1. Figure 1 construction management: concrete construction bridge construction:How to become a bridge engineer the procedure of analyzing of T Beams Compute T=As*fy determining if a > or < hf by calculating AC=T/0.85*fc', if AC>Area of flange(hf*bf) then a is more than hf if a<hf As*fy=0.85*fc'*a*bf a=(As*fy)/(0.85*fc'*bf) if a>hf AC=AC by flange +AC by web AC by web=(AC-AC by flange) bw*(a-hf)=(AC-AC by flange) a=(AC-AC by flange)/(bw)+hf AC by flange=0.85*fc'*hf*bf for this case we need to de...
Read more

Non-Working Pile Testing and Static Compression Load Test

By -
Image
Purpose of Non-Working Piles Non-working piles are constructed at the beginning of the project to evaluate soil bearing capacity and determine the maximum settlement. These piles are loaded up to twice the working load. Based on the test results, the pile length may be adjusted. If the measured settlement exceeds the allowable limit, the pile length will be increased accordingly. construction management: concrete construction bridge construction:How to become a bridge engineer Static Compression Load Test The static compression load test provides insight into how the pile responds to loading. Settlement is measured at different load increments to determine the appropriate pile design length. The maximum applied load is typically twice the working load (2 × working load) or as specified by local standards. The following steps outline the testing procedure: A. Steel Cage Preparation The steel reinforcement cage is prepared in accordance with the approved shop drawings. ...
Read more

shear key

By -
Image
The purpose of the shear key is to improve the bonding between concrete surfaces. The shear key is usually made at the construction joint location to enhance the adhesion between the old and new casted concrete. Construction joint is considered a weak point because the concrete is not casted at one time, in large structures where the concrete can not be casted for some members at one time, such as a slab. We usually divide the slab and cast it at many stages. The large quantities of concrete will make it impossible to cast the slab at one time. Therefore, the casting will be done in several stages. The location of the construction joint should be selected carefully. Construction joint shall be made at the location of minimum shear and moment.  The bonding between old and new concrete can be improved by roughening the old concrete and forming a shear key on the concrete surface while the concrete is still plastic. The shear key can be formed by pressing timber into the concrete surf...
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

Flexible Pavement Layers and Their Components

By -
Image
Flexible pavement is composed of several layers that work together to distribute traffic loads to the subgrade. These layers are generally divided into aggregate layers and asphaltic layers , each serving a specific function to ensure strength, stability, and durability of the pavement. Aggregate Layers The aggregate portion of the pavement usually consists of the subbase , roadbase , and sometimes a wet mix macadam (WMM) layer. These layers play a crucial role in supporting the asphaltic layers above and ensuring that loads are transferred smoothly to the subgrade without excessive settlement or deformation. 1-Subbase The subbase is a mixture of fine and coarse aggregates placed directly above the existing subgrade. Its main purpose is to improve the pavement’s overall bearing capacity, minimize settlement, and provide a uniform surface for the upper layers. Typically, the subbase should have a California Bearing Ratio (CBR) of not less than 30. The minimum thickness is usually abo...
Read more

Pressure Reducing (or Pressure Breaking) Valves

By -
Image
A pressure reducing valve (PRV) , also known as a pressure breaking valve , is used to reduce and regulate fluid pressure to a specific, safe value in order to protect downstream equipment and piping systems. These valves are commonly used in water supply and hydraulic systems , where the inlet pressure may be higher than the pressure rating of the downstream components. In such cases, the pressure reducing valve ensures that the downstream pressure remains within the allowable range of the equipment and pipes, preventing possible damage or system failure. The figure below shows the main components of a pressure reducing valve, which typically include a diaphragm , spring , inlet , and outlet . The valve is set to a preset pressure value . When the inlet pressure exceeds this preset limit, the diaphragm is pushed upward, compressing the spring and closing the valve partially to restrict flow — thereby reducing the downstream pressure. Conversely, when the inlet pressure drops bel...
Read more

ACI Reinforcement Limits Explained: Minimum and Maximum Steel Area for Concrete Columns

By -
Image
According to ACI, the area of longitudinal reinforcement for a column shall not be less than 0.01·Ag and shall not exceed 0.08·Ag (Ag = gross sectional area of the column). The minimum reinforcement requirement ensures the column has sufficient reinforcement to resist bending that may arise from misalignment during construction or other causes, and to control shrinkage and creep under sustained load. The maximum reinforcement limit of 8% is intended to prevent overly brittle behavior and sudden failure without warning. A very large amount of reinforcement also causes congestion, making proper placement of bars, ties, and concrete consolidation impractical. The 8% limitation applies to the total longitudinal area at a section — the designer must therefore consider column length, bar lengths, splice locations, and lap splice practices when checking the maximum reinforcement criterion. Example (60 cm × 40 cm column) Gross area: Ag = 600 mm × 400 mm = 240,000 mm². Minimum longitudinal...
Read more