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- How to calculate the maximum deflection of a simply supported beam with spring support in the middle when it's subjected to uniformly distributed load ?

Hi,

I'd like to calculate the maximum deflection of a simply supported beam with spring support in the middle and UDL (uniformly distributed load) acting on the whole beam:

Here's my derivation, starting from the known formula for maximum deflection of a simply supported beam with UDL and no spring: $$\delta_{b}=\frac{5 \left( \frac{F}{L} \right) L^{4}}{384EI}$$ $$k_{b}=\frac{F}{\delta_{b}}=\frac{384EI}{5L^{3}}$$ $$k_{t}=k_{b}+k_{s}=\frac{384EI}{5L^{3}}+k_{s}$$ $$\delta_{t}=\frac{F}{k_{t}}=\frac{5FL^{3}}{384EI+5kL^{3}}$$

The problem is that when I substitute the data: ##F=2000 \ N##, ##L=500 \ mm##, ##E=210 \ GPa##, ##I=\frac{a^{4}}{12}=\frac{12^{4}}{12}=1728 \ mm^{4}##, ##k=100 \ \frac{N}{mm}##

the result is: $$\delta_{t}=\frac{5 \cdot 2000 \cdot 500^{3}}{384 \cdot 210000 \cdot 1728+5 \cdot 100 \cdot 500^{3}}=6.1928 \ mm$$

while from FEA, I get: ##5.1796 \ mm## and I believe this result is correct. What's wrong with my formula ?

Interestingly, I got a very good agreement for the same case with a concentrated force in the middle instead of UDL (using the same approach, just a different base formula for the deflection of the beam). Does it mean that the ##q=\frac{F}{L}## conversion is incorrect here ? How should I treat it then ?

I'd like to calculate the maximum deflection of a simply supported beam with spring support in the middle and UDL (uniformly distributed load) acting on the whole beam:

Here's my derivation, starting from the known formula for maximum deflection of a simply supported beam with UDL and no spring: $$\delta_{b}=\frac{5 \left( \frac{F}{L} \right) L^{4}}{384EI}$$ $$k_{b}=\frac{F}{\delta_{b}}=\frac{384EI}{5L^{3}}$$ $$k_{t}=k_{b}+k_{s}=\frac{384EI}{5L^{3}}+k_{s}$$ $$\delta_{t}=\frac{F}{k_{t}}=\frac{5FL^{3}}{384EI+5kL^{3}}$$

The problem is that when I substitute the data: ##F=2000 \ N##, ##L=500 \ mm##, ##E=210 \ GPa##, ##I=\frac{a^{4}}{12}=\frac{12^{4}}{12}=1728 \ mm^{4}##, ##k=100 \ \frac{N}{mm}##

the result is: $$\delta_{t}=\frac{5 \cdot 2000 \cdot 500^{3}}{384 \cdot 210000 \cdot 1728+5 \cdot 100 \cdot 500^{3}}=6.1928 \ mm$$

while from FEA, I get: ##5.1796 \ mm## and I believe this result is correct. What's wrong with my formula ?

Interestingly, I got a very good agreement for the same case with a concentrated force in the middle instead of UDL (using the same approach, just a different base formula for the deflection of the beam). Does it mean that the ##q=\frac{F}{L}## conversion is incorrect here ? How should I treat it then ?

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