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changed title, given structure isn't a cantilever
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Wasabi
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Deformation of a two segment cantileverfixed-fixed beam with perfectly plastic yield stresses

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Wasabi
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Here's the setup:

Cantilever set-up

The problem asks to find the relationship between the force $P$ and the displacement of B ($u_B$). I have done this correctly:

$$P = \left(\dfrac{E_1A_1}{L_1}+\dfrac{E_2A_2}{L_2}\right)u_B$$

Now imagine that members AB and BC both have yield stresses ($\sigma_{Y_1}$ and $\sigma_{Y_2}$) and are perfectly plastic after yield.

I need to find the relationship between uB$u_B$ and P$P$ in the following cases:

  1. No member yields - ($0 < u_B < \sigma_{Y_1}$) & ($0 < P < P_{y_1}$)
  2. Member AB yields, but not member BC - ($u_{Y_1} < u_B < u_{Y_2}$) & ($P_{y,1} < P < P_{y,2}$)
  3. Both members yield - ($u_B > u_{Y_2}$) & ($P > P_{y,2}$)

I'm having trouble finding explicit expressions for $P_{y_{1/2}}$ and $u_{Y_{1/2}}$. I'm not quite sure how to relate yield stress to elongation and displacement of the members.

To be honest, this problem has tied my brain in a knot and I believe my conceptual understanding (or misunderstanding) of the problem is limiting my ability to solve it mathematically. Any help at all would be greatly appreciated!

Edit: Here is the exact problem Exact Problem

Here's the setup:

Cantilever set-up

The problem asks to find the relationship between the force $P$ and the displacement of B ($u_B$). I have done this correctly:

$$P = \left(\dfrac{E_1A_1}{L_1}+\dfrac{E_2A_2}{L_2}\right)u_B$$

Now imagine that members AB and BC both have yield stresses ($\sigma_{Y_1}$ and $\sigma_{Y_2}$) and are perfectly plastic after yield.

I need to find the relationship between uB and P in the following cases:

  1. No member yields - ($0 < u_B < \sigma_{Y_1}$) & ($0 < P < P_{y_1}$)
  2. Member AB yields, but not member BC - ($u_{Y_1} < u_B < u_{Y_2}$) & ($P_{y,1} < P < P_{y,2}$)
  3. Both members yield - ($u_B > u_{Y_2}$) & ($P > P_{y,2}$)

I'm having trouble finding explicit expressions for $P_{y_{1/2}}$ and $u_{Y_{1/2}}$. I'm not quite sure how to relate yield stress to elongation and displacement of the members.

To be honest, this problem has tied my brain in a knot and I believe my conceptual understanding (or misunderstanding) of the problem is limiting my ability to solve it mathematically. Any help at all would be greatly appreciated!

Edit: Here is the exact problem Exact Problem

Here's the setup:

Cantilever set-up

The problem asks to find the relationship between the force $P$ and the displacement of B ($u_B$). I have done this correctly:

$$P = \left(\dfrac{E_1A_1}{L_1}+\dfrac{E_2A_2}{L_2}\right)u_B$$

Now imagine that members AB and BC both have yield stresses ($\sigma_{Y_1}$ and $\sigma_{Y_2}$) and are perfectly plastic after yield.

I need to find the relationship between $u_B$ and $P$ in the following cases:

  1. No member yields - ($0 < u_B < \sigma_{Y_1}$) & ($0 < P < P_{y_1}$)
  2. Member AB yields, but not member BC - ($u_{Y_1} < u_B < u_{Y_2}$) & ($P_{y,1} < P < P_{y,2}$)
  3. Both members yield - ($u_B > u_{Y_2}$) & ($P > P_{y,2}$)

I'm having trouble finding explicit expressions for $P_{y_{1/2}}$ and $u_{Y_{1/2}}$. I'm not quite sure how to relate yield stress to elongation and displacement of the members.

To be honest, this problem has tied my brain in a knot and I believe my conceptual understanding (or misunderstanding) of the problem is limiting my ability to solve it mathematically. Any help at all would be greatly appreciated!

Edit: Here is the exact problem Exact Problem

added 44 characters in body
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Here's the setup:

Cantilever set-up

The problem asks to find the relationship between the force $P$ and the displacement of B ($u_B$). I have done this correctly:

$$P = \left(\dfrac{E_1A_1}{L_1}+\dfrac{E_2A_2}{L_2}\right)u_B$$

Now imagine that members AB and BC both have yield stresses ($\sigma_{Y_1}$ and $\sigma_{Y_2}$) and are perfectly plastic after yield.

I need to find the relationship between uB and P in the following cases:

  1. No member yields - ($0 < u_B < \sigma_{Y_1}$) & ($0 < P < P_{y_1}$)
  2. Member AB yields, but not member BC - ($u_{Y_1} < u_B < u_{Y_2}$) & ($P_{y,1} < P < P_{y,2}$)
  3. Both members yield - ($u_B > u_{Y_2}$) & ($P > P_{y,2}$)

I'm having trouble finding explicit expressions for $P_{y_{1/2}}$ and $u_{Y_{1/2}}$. I'm not quite sure how to relate yield stress to elongation and displacement of the members.

To be honest, this problem has tied my brain in a knot and I believe my conceptual understanding (or misunderstanding) of the problem is limiting my ability to solve it mathematically. Any help at all would be greatly appreciated!

Edit: Here is the exact problem Exact Problem

Here's the setup:

Cantilever set-up

The problem asks to find the relationship between the force $P$ and the displacement of B ($u_B$). I have done this correctly:

$$P = \left(\dfrac{E_1A_1}{L_1}+\dfrac{E_2A_2}{L_2}\right)u_B$$

Now imagine that members AB and BC both have yield stresses ($\sigma_{Y_1}$ and $\sigma_{Y_2}$) and are perfectly plastic after yield.

I need to find the relationship between uB and P in the following cases:

  1. No member yields - ($0 < u_B < \sigma_{Y_1}$) & ($0 < P < P_{y_1}$)
  2. Member AB yields, but not member BC - ($u_{Y_1} < u_B < u_{Y_2}$) & ($P_{y,1} < P < P_{y,2}$)
  3. Both members yield - ($u_B > u_{Y_2}$) & ($P > P_{y,2}$)

I'm having trouble finding explicit expressions for $P_{y_{1/2}}$ and $u_{Y_{1/2}}$. I'm not quite sure how to relate yield stress to elongation and displacement of the members.

To be honest, this problem has tied my brain in a knot and I believe my conceptual understanding (or misunderstanding) of the problem is limiting my ability to solve it mathematically.

Here's the setup:

Cantilever set-up

The problem asks to find the relationship between the force $P$ and the displacement of B ($u_B$). I have done this correctly:

$$P = \left(\dfrac{E_1A_1}{L_1}+\dfrac{E_2A_2}{L_2}\right)u_B$$

Now imagine that members AB and BC both have yield stresses ($\sigma_{Y_1}$ and $\sigma_{Y_2}$) and are perfectly plastic after yield.

I need to find the relationship between uB and P in the following cases:

  1. No member yields - ($0 < u_B < \sigma_{Y_1}$) & ($0 < P < P_{y_1}$)
  2. Member AB yields, but not member BC - ($u_{Y_1} < u_B < u_{Y_2}$) & ($P_{y,1} < P < P_{y,2}$)
  3. Both members yield - ($u_B > u_{Y_2}$) & ($P > P_{y,2}$)

I'm having trouble finding explicit expressions for $P_{y_{1/2}}$ and $u_{Y_{1/2}}$. I'm not quite sure how to relate yield stress to elongation and displacement of the members.

To be honest, this problem has tied my brain in a knot and I believe my conceptual understanding (or misunderstanding) of the problem is limiting my ability to solve it mathematically. Any help at all would be greatly appreciated!

Edit: Here is the exact problem Exact Problem

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