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Q) Determine the reactions, sectional forces, and maximum bending moment of the following structure,

and construct the shear force diagram (SFD) and bending moment diagram (BMD).

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Answer)

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1. Reactions (R_A, R_B, H_A)

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sol1 R.A L * w L * 2 / (L 3 / * - 0 ≡ R.A w L * 2 / - R.B + 0 ≡ H.A 0 + 31 mat R.A R.B H.A 31 mat Solve : R.A L w * 6 / ≡ R.B L w * 3 / ≡ H.A 0 ≡ 31 sys

sol1 Assign L w * 6 / L w * 3 / 031 sys

∑M_A=0(+;⟳)

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∑F_y=0(+;↑)

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∑F_x=0(+;→)

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R.A L w * 6 /

R.B L w * 3 /

H.A 0

2. Internal Forces

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2.1 Shear Force

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x V R.A w x * L / x * 12 / * - : L w * 6 / w x * L / x * 12 / * -

x V factor Maxima w 3 x 2^* L 2^- (* 6 L * / -

2.2 Bending Moment

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x M R.A x * w x * L / x * 12 / * (x 3 / * - : L w * 6 / x * w x * L / x * 12 / * x 3 / * -

x M factor Maxima w x * x L - (* x L + (* 6 L * / -

3. Maximum Bending Moment

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3.1 Location of Maximum Bending Moment

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sol2 x V 0 ≡ x Solve : x L 3 sqrt / - ≡ x L 3 sqrt / ≡ 21 sys

sol2 2 el Assign L 3 sqrt /

x.max x : L 3 sqrt /

3.2 Calculation of Maximum Moment (M_max)

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M.max x.max M : L w * 6 / L 3 sqrt / * w L 3 sqrt / * L / L 3 sqrt / * 12 / * L 3 sqrt / 3 / * -

M.max factor Maxima L 2^w * 352 /^/

352 /^15.5885

93 sqrt * 15.5885

4. SFD, BMD

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x.max x Clear 1

x V w L 2^3 x 2^* - (* 6 L * /

x M w x * L x - (* L x + (* 6 L * /

x V w 1 ≡ L 1 ≡ 2 1 sys at x M w 1 ≡ L 1 ≡ 2 1 sys at 0 0 + 800 1 4 mat 3 1 sys