GATE CE 2001

\begin{aligned} I&=\int_{0}^{\pi /4}\cos ^{2}xdx \\ &=\int_{0}^{\pi /4}\frac{1+\cos 2x}{2}dx \\ &=\int_{0}^{\pi /4}\frac{dX}{2}+\int_{0}^{\pi /4}\frac{\cos 2x}{2}dx \\ &=\frac{1}{2}\left [ x \right ]_{0}^{\pi /4}+\frac{1}{2}\left [ \frac{\sin 2x}{2} \right ]_{0}^{\pi /4} \\ &=\frac{\pi }{8}+\frac{1}{4}\left [ \sin \frac{\pi }{2}-\sin x \right ] \\ &=\frac{\pi }{8}+\frac{1}{4}\times 1 =\frac{\pi }{8}+\frac{1}{4} \end{aligned}

\lim_{x\rightarrow \frac{\pi }{2}}f\left ( x \right )=\lim_{x\rightarrow \frac{\pi }{2}}\sin x=1

Degree of static indeterminacy,
\begin{aligned} N_{s} &=3 m+r_{e}-3 j-r_{r} \\ &=3 \times 5+(3+2+2)-3 \times 6-0 \\ &=4 \end{aligned}
Degree of kinematic indeterminacy,
\begin{aligned} N_{k} &=3 j-r_{e}-m \\ &=3 \times 6-(3+2+2)-5=6 \\ N_{s} &=4 \text { and } N_{k}=6 \end{aligned}

\begin{aligned} \mathrm{R}+\mathrm{V}&=30 &\ldots(i)\\ \text { Now, } \mathrm{V} &\times 2-20 \times 3+10 \times 1=0 \\ \mathrm{V}&=\frac{60-10}{2}=25 \mathrm{kN}\\ \therefore \quad \mathrm{R}&=5 \mathrm{kN}\\ \mathrm{BM} \text { at }\quad X&=25 \times 1-20 \times 2=-15 \mathrm{kNm} \end{aligned}

As temperature increases the bar AD will tend to elongate but joint B and D will offer resistance.
Hence bar BD will be in compression.

Since, there is no bending moment in vertical leg so there will be no bending in member AB. And to maintain angle between AB and BC at joint 8, frame will sway as given below

The most common type of rivet patterns are chain riveting and diamond riveting. Staggered pattern in option (A) yields more net area of the section and because of this reaon this pattern is most suitable for tension members. Staggered and diamond pattern are better as compared to the chain pattern.