Solid Mechanics Stiffness Question

In summary, the conversation discusses the construction of a concrete post using reinforced steel rods. The goal is to determine the necessary number of steel rods to achieve an axial stiffness of at least 2000kN/mm. The equation used to solve this problem is k=F/EA, where k is the stiffness, F is the force, and EA is the product of the modulus of elasticity and the cross-sectional area. Through strain compatibility, the final equation is n = (4D^2)/(5d^2), which should equal 80 but does not. The individual discussing the problem is unsure of where they went wrong in their calculations.
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Homework Statement



A 0.5m high and 100mm diameter post is to be made from concrete reinforced by 10mm diameter steel rods. How many steel rods should be used if the axial stiffness of the post is to be at least 2000kN/mm?

Homework Equations



n = number of steel rods
D = diamater of the post
d = Diameter of the steel rods

A Concrete= [tex]\stackrel{piD 2}{4}[/tex] - n [tex]\stackrel{pid 2}{4}[/tex]
A Steel = n [tex]\stackrel{pid 2}{4}[/tex]k (Stiffness) = F/(EA)

The Attempt at a Solution



Strain compatibility is assumed
[tex]\frac{E SteelA Steel}{L}[/tex] = [tex]\frac{E ConcreteA Concreete}{L}[/tex]

Substitute in Areas, and cancel where possible, I am left with n = [tex]\frac{4D2}{5d2}[/tex] which = 80, which is not right but i am struggling to see what I have done wrong.
 
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  • #2
In your equ k=F/EA, what are the units of k on the LHS and what are the units of F/EA on the RHS? Isn't there something missing?
 

1. What is the definition of stiffness in solid mechanics?

In solid mechanics, stiffness refers to the resistance of a material to deformation or change in shape under an applied force. It is a measure of how difficult it is to stretch, bend, or compress a material.

2. How is stiffness measured in solid mechanics?

Stiffness is typically measured using the modulus of elasticity, also known as Young's modulus. This is a material property that describes the ratio of stress (force per unit area) to strain (change in length per unit length) in a material.

3. What factors affect the stiffness of a material?

The stiffness of a material is affected by its composition, microstructure, and temperature. Generally, materials with higher modulus of elasticity values are stiffer, while materials with lower modulus of elasticity values are more flexible.

4. How does stiffness impact the behavior of structures?

Stiffness plays a crucial role in determining the behavior of structures under external loads. A stiffer material or structure will experience less deformation and more stress under a given load, while a less stiff material or structure will experience more deformation and less stress.

5. Can stiffness be changed in a material?

Yes, stiffness can be changed in a material by altering its composition, microstructure, or temperature. For example, adding reinforcement materials or changing the processing method can increase the stiffness of a material.

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