Exact Analysis of Indeterminate Structures: Force Methods - Examples

A propped cantilever is statically indeterminate to the first degree. Identify one convenient redundant.

At the released redundant coordinate, real loads cause $\Delta^L=-8.0\text{ mm}$ and a unit redundant causes flexibility $f=0.040\text{ mm/kN}$. The original support requires zero displacement. Find $X$.

Using the same system, suppose the calculation gives $X=-35\text{ kN}$ while the assumed redundant arrow was upward. Interpret the result.

Over a $3.0\text{ m}$ prismatic beam segment, unit-load moment diagrams are constant at $m_i=1.5\text{ m}$ and $m_j=2.0\text{ m}$. Let $EI=60000\text{ kN}\cdot\text{m}^2$. Find $f_{ij}$.

A calculation gives $f_{12}=0.075\text{ mm/kN}$ and $f_{21}=0.091\text{ mm/kN}$ for a stable linear-elastic model using the same generalized coordinates. What should be done?

A structure has compatibility equations

with displacement in mm and flexibility in mm/kN. Solve for $X_1$ and $X_2$.

At a redundant support, real loads cause $+4.0\text{ mm}$ displacement in the positive redundant direction, while the support itself settles $+2.0\text{ mm}$ in that same direction. Flexibility is $f=0.030\text{ mm/kN}$. Find $X$ from $\Delta^L+fX=\Delta^{specified}$.

A released tie of length $L=6.0\text{ m}$ would freely expand under $\alpha=12\times10^{-6}/^\circ\text{C}$ and $\Delta T=30^\circ\text{C}$. Find the free thermal elongation to be included in compatibility.

A two-degree indeterminate frame is analyzed by releasing two moments, but the released primary structure becomes a lateral mechanism. Is the redundant set acceptable?

A symmetric beam and support system carries a symmetric loading. What qualitative simplification should be checked before solving multiple redundants?

After restoring redundants, a continuous beam satisfies all force-equilibrium equations but the vertical displacement at an original roller support is calculated as $5\text{ mm}$ when no settlement was prescribed. Is the solution acceptable?