Deflection of a Beam: Solving for vB using the Double-Integration Method

In summary, the conversation discusses using the double-integration method to determine the deflection of a simply supported steel beam at point B. The given parameters are E = 200 GPa, I = 129 × 10^6 mm4, L = 4 m, P = 60 kN, and w= 40 kN/m. The conversation also mentions checking the equation for bending moment in the beam and splitting the problem into calculating the bending moment due to the central point load and the distributed load separately. It is suggested to draw the shear and bending moment diagrams for the beam to avoid confusion.
  • #1
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Homework Statement


For the simply supported steel beam [E = 200 GPa; I = 129 × 10^6 mm4], use the double-integration method to determine the deflection vB at B. Assume L = 4 m, P = 60 kN, and w= 40 kN/m.

Can someone please help, when I insert x=2m into the equation i derived for M, and then insert M into equation 1, I do not get the correct answer (-8.27 mm).

Homework Equations


E.I.y''=M (eq. 1)
M=F.d (eq. 2)

The Attempt at a Solution


[/B]
 

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  • #2
Check your equation for the bending moment in the beam. I would split the problem up into calculating the BM due to the central point load plus the BM due to the distributed load.

In order to avoid confusion, draw the shear and bending moment diagrams for this beam. Because the beam is symmetrically loaded, these can almost be done by inspection.
 

FAQ: Deflection of a Beam: Solving for vB using the Double-Integration Method

1. What is the formula for calculating deflection of a beam?

The formula for calculating deflection of a beam is given by: Deflection = (F x L^3) / (3 x E x I), where F is the applied force, L is the length of the beam, E is the modulus of elasticity, and I is the moment of inertia.

2. How does the shape of a beam affect its deflection?

The shape of a beam plays a significant role in its deflection. A beam with a larger cross-sectional area will have a lower deflection compared to a beam with a smaller cross-sectional area. Similarly, a beam with a larger moment of inertia will have a lower deflection compared to a beam with a smaller moment of inertia.

3. What is the significance of deflection in structural engineering?

Deflection is an important consideration in structural engineering as it determines the stability and strength of a structure. Too much deflection can lead to structural failure, while too little can result in an unnecessarily rigid and heavy structure. Engineers must carefully calculate and consider deflection in their designs to ensure the safety and efficiency of a structure.

4. How does the material of a beam affect its deflection?

The material of a beam also plays a role in its deflection. Materials with higher modulus of elasticity, such as steel, will have a lower deflection compared to materials with a lower modulus of elasticity, such as wood. Additionally, the strength and stiffness of a material can also affect its deflection.

5. Are there any limitations to the formula for calculating deflection of a beam?

Yes, there are limitations to the formula for calculating deflection of a beam. It assumes that the beam is loaded within its elastic limit and that its cross-sectional area remains constant. If these assumptions are not met, the calculated deflection may not accurately reflect the actual deflection of the beam. Additionally, the formula does not account for factors such as temperature changes or creep, which can also affect the deflection of a beam.

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