Create a 1–2 minute visual explanation for this structural mechanics problem:
A simply supported beam AE of length L has two equal point loads P applied at one-quarter and three-quarter spans (points B and D), and an additional force F applied at the midspan (point C). The beam has constant flexural rigidity EI.
The goal is to find the value of F such that the vertical deflection at point C becomes zero.
Please:
Clearly label the beam, supports (A and E), and the loads at B, D, and C.
Show how to apply the superposition principle to compute deflection at point C due to each load.
Use visual diagrams to help explain how each load affects point C.
Show the final equation summing the deflections to zero and solving for F.
Focus on the visual logic of why F is needed to cancel out the downward deflection caused by the two P loads.---**Textual Information:**
有一橫樑 $AE$ 受到兩個集中力 $P$ 作用，如下圖所示，樑斷面之撓曲剛度為常數 $EI$:
There is a beam $AE$ with constant flexural rigidity as $EI$ and a series loading.
此時若增加集中力 $F$ 欲使樑中 $C$ 點回到中線（抵銷垂直位移），則剛集中力大小為何？
If the vertical deflection of the beam at midspan (i.e., point $C$) is to be zero, determine the magnitude of force $F$.
**Chart/Diagram Description:**
* **Type:** Structural diagram of a beam under loading.
* **Main Elements:**
* **Beam:** A horizontal beam labeled $AE$.
* **Supports:** A pin support is located at point $A$ on the left end. A roller support is located at point $E$ on the right end.
* **Points:** Points $A$, $B$, $C$, $D$, and $E$ are marked along the beam from left to right. Point $C$ is located at the midspan.
* **Loads:**
* A downward vertical point force $P$ is applied at point $B$.
* A downward vertical point force $P$ is applied at point $D$.
* An upward vertical point force labeled $F=?$ is applied at point $C$.
* **Dimensions:** The beam is divided into four equal segments by points $A, B, C, D, E$. The lengths of segments $AB$, $BC$, $CD$, and $DE$ are all labeled as $L/4$. The total length of the beam is $L$.
* **Annotations:** The flexural rigidity is mentioned as $EI$ in the text. The unknown force is labeled as $F=?$.
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We have a simply supported beam AE of length L. The beam is supported by a pin at point A and a roller at point E. Two equal downward loads P are applied at points B and D, located at one quarter and three quarter spans respectively. An unknown upward force F is applied at the midspan point C. Our goal is to find the value of F that makes the vertical deflection at point C equal to zero.
To solve this problem, we use the superposition principle. The total deflection at point C is the sum of deflections caused by each load acting independently. We analyze three separate cases: first, the effect of load P at point B only; second, the effect of load P at point D only; and third, the effect of force F at point C only. Each load causes the beam to deflect, and we can see the deflection curves for each case. The green curves show how each individual load affects the beam shape.
Now let's calculate the specific deflections using standard beam formulas. For a simply supported beam, when load P is applied at one quarter span, the deflection at the center is eleven P L cubed over three hundred eighty four E I. Due to symmetry, the load P at three quarter span produces the same deflection at center. The total downward deflection from both P loads is eleven P L cubed over one hundred ninety two E I. The upward force F at the center produces an upward deflection of F L cubed over forty eight E I. For zero deflection at point C, these must balance exactly.
Now we solve for the required force F. Setting the total deflection at C equal to zero, we get the equilibrium equation. The downward deflection from the P loads equals eleven P L cubed over one hundred ninety two E I, and this must equal the upward deflection from force F. Solving this equation, we find that F equals eleven P over four. The top diagram shows the beam deflected downward by the P loads only. The bottom diagram shows the corrected beam with force F applied, resulting in zero deflection at point C. This demonstrates how the upward force F exactly cancels the downward deflection caused by the two P loads.
To summarize what we have learned: We applied the superposition principle to analyze beam deflection by considering each load separately. We calculated individual deflections using standard beam formulas and found that a force F equal to eleven P over four creates zero deflection at the midspan. This demonstrates how an upward force can precisely cancel downward deflections. This systematic approach applies to any simply supported beam analysis problem.