Statics 3.6 – Equilibrium of Truss Members Connected to a Gusset Plate | Hibbeler 14th Edition

The gusset plate is subjected to the forces of three members. Determine the tension force in member C and its angle θ for equilibrium. The forces are concurrent at point O. Take F=8 kN.

Engineering Mechanics: Statics 14th Edition by RC Hibbeler Problem 3-5 Equilibrium of Truss Members Connected in a Gusset Plate

Solution:

Free-body diagram:

Free-body diagram of Problem 3.6 of Engineering Mechanics Statics by RC Hibbeler 14th Edition: The equilibrium of truss members connected to a gusset plate

Equations of Equilibrium:

Taking the sum of forces in the x-direction:

Tcos⁡β−45(8)=0Tcos⁡β=45(8)Tcos⁡β=6.4(1)\begin{aligned} T \cos \beta-\frac{4}{5}(8) & = 0 & & \\ T \cos \beta& = \frac{4}{5}(8) & &\\ T \cos \beta & = 6.4 \qquad \qquad & & (1)\\ \end{aligned}

Taking the sum of forces in the y-direction:

9−35(8)−Tsin⁡β=0Tsin⁡β=9−35(8)Tsin⁡β=4.2(2)\begin{aligned} 9-\frac{3}{5}(8)-T \sin \beta & = 0 & &\\ T \sin \beta & =9-\frac{3}{5}(8) & & \\ T \sin \beta & =4.2 & &\qquad \qquad (2) \end{aligned}

Equation (2) divided by equation (1) to solve for angle β.

Tsin⁡βTcos⁡β=4.26.4tan⁡β=0.65625β=tan⁡−1(0.65625)β=33.27°\begin{aligned} \dfrac{T \sin \beta}{T \cos \beta} & = \frac{4.2}{6.4} \\ \tan \beta & = 0.65625 \\ \beta & =\tan ^{-1}(0.65625) \\ \beta & =33.27 \degree \end{aligned}

Substitute the solved value of the angle β to equation (1) to solve for T.

Tcos⁡β=6.4T=6.4cos⁡βT=6.4cos⁡33.27°T=7.65 kN\begin{aligned} T \cos \beta & =6.4 \\ T & = \dfrac{6.4}{\cos \beta}\\ T & = \dfrac{6.4}{\cos 33.27 \degree}\\ T & =7.65 \ \text{kN} \end{aligned}

Solve for the value of the unknown angle θ:

θ=β+tan⁡−1(34)θ=33.27°+36.87°θ=70.14°\begin{aligned} \theta & = \beta +\tan ^{-1} \left( \frac{3}{4}\right) \\ \theta & = 33.27 \degree + 36.87 \degree \\ \theta & =70.14\degree \end{aligned}

Therefore, the tension force in member C is 7.65 kN and its angle θ is 70.14°.