### Lamis Theorem

• Recount and apply Lami’s theorem.

### Solved Example:

#### 24-5-01

Two steel truss members, AC and BC, each having cross sectional area of 100 mm$^2$, are subjected to a horizontal force F as shown in figure. All the joints are hinged. If F = 1 kN, the magnitude of the vertical reaction force developed at the point B in kN is: (GATE ME 2012) Solution:
$\dfrac {F}{\sin 105^\circ}=\dfrac {T_{2}}{\sin 120^\circ}=\dfrac {T_{1}}{\sin 135^\circ}$ $\dfrac {T_{1}}{\sin 135^\circ}=\dfrac {F}{\sin 105^\circ}=\dfrac {1}{\sin 105^\circ}$ $T_{1}=0.7320kN$ $R_{NT1} = T_{1}\cos 30^\circ = 0.73205\times \cos 30^\circ = 0.634\ kN$

### Solved Example:

#### 24-5-02

The maximum force F is kN that can be applied at C such that the axial stress in any of the truss members DOES NOT exceed 100 MPa is: (GATE ME 2012)

Solution:
$\dfrac {F}{\sin 105^{\circ }}=\dfrac {T_{2}}{\sin 120^{0}}$ $T_{2}=\dfrac {\sin 120^{0}}{\sin 135^{\circ }}\times F$ $T_{1}=\left( 0.73205\right)F$ $T_{2} > T_{1}$ $\sigma =100\ MPa$ $F=\sigma \times A_{1}$ $F_{\max }=\sigma _{\max }\times A_{1}$ $T_{2}=100\times 100$ $0.8965F=100\times 100$ $F_{1}=\dfrac {100\times 100}{0.8965} =11154.5\ N =11.15\ kN$

### Solved Example:

#### 24-5-03

A rope is stretched between two rigid walls 40 meters apart. At the midpoint, a load of 100 N was placed that caused it to sag 5 meters. Compute the approximate tension in the rope.

Solution:
$\theta =\tan ^{-1}\left( \dfrac {5}{20}\right) =14.03^\circ$ $\alpha =\theta +90^\circ =104.03^\circ$ $\beta =\alpha = 104.03^\circ$ $\gamma =360-\alpha -\beta =151.93^\circ$ Using Lami's theorem, $\dfrac {F_{1}}{\sin \alpha }=\dfrac {F_{2}}{\sin \beta }=\dfrac {F_{3}}{\sin \gamma }$ $\dfrac {F_{1}}{\sin 104.03^\circ}=\dfrac {100}{\sin 151.93^\circ}$ $F_{1}=206.16\ N$