seat work
1. a beam supported on three reactions as shown consists of two segment joined at a frictionless hinge at which the bending moment is zero.

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Hinged Beam Analysis", "Static Equilibrium", "Beam Reactions", "Shear and Moment Diagrams" ], "new_concepts": [], "current_concepts": [ "Hinged Beam Analysis", "Static Equilibrium", "Beam Reactions", "Shear and Moment Diagrams" ] } ## Analyze the beam segments and equilibrium of segment AC $$ \begin{aligned} &\text{Hinge at } D \implies M_D = 0\ &\text{Segment } AD \text{ has length } L_{AD} = 4\text{ m} + 1\text{ m} = 5\text{ m}\ &\text{Segment } DE \text{ has length } L_{DE} = 5\text{ m}\ &\text{For segment } AD \text{ (from } A \text{ to } D\text{):}\ &\quad \text{UDL of } 20\text{ kN/m} \text{ over } 2\text{ m from } A \text{ to } B\ &\quad \text{Point load of } 40\text{ kN} \text{ at } B\ &\quad \text{Reaction } R_1 \text{ at } A\ &\quad \text{Reaction } R_2 \text{ at } C \text{ (at } 4\text{ m from } A\text{)}\ &\quad \text{Internal shear force } V_D \text{ at the hinge } D \text{ (at } 5\text{ m from } A\text{)}\ &\sum M_C = 0 \text{ for segment } AD:\ &\quad -R_1(4) + (20 \times 2)(3) + 40(2) + V_D(1) = 0 \implies -4R_1 + 120 + 80 + V_D = 0 \implies 4R_1 - V_D = 200\ &\sum M_D = 0 \text{ for segment } AD:\ &\quad -R_1(5) - R_2(1) + (20 \times 2)(4) + 40(3) = 0 \implies 5R_1 + R_2 = 280 \end{aligned} $$ ## Analyze segment DE and solve for reactions $$ \begin{aligned} &\text{For segment } DE \text{ (from } D \text{ to } E\text{):}\ &\quad \text{Length } L_{DE} = 5\text{ m}\ &\quad \text{UDL of } 20\text{ kN/m} \text{ over } 4\text{ m (from } 1\text{ m to } 5\text{ m from } D\text{)}\ &\quad \text{Reaction } R_3 \text{ at } E \text{ (at } 5\text{ m from } D\text{)}\ &\quad \text{Hinge force } V_D \text{ acting upwards on } DE\ &\sum M_E = 0 \text{ for segment } DE:\ &\quad -V_D(5) + (20 \times 4)(2) = 0 \implies 5V_D = 160 \implies V_D = 32\text{ kN}\ &\text{Substitute } V_D = 32\text{ kN} \text{ into segment } AD \text{ equations}:\ &\quad 4R_1 - 32 = 200 \implies 4R_1 = 232 \implies R_1 = 58\text{ kN}\ &\quad 5(58) + R_2 = 280 \implies 290 + R_2 = 280 \implies R_2 = -10\text{ kN}\ &\sum F_y = 0 \text{ for segment } DE:\ &\quad V_D + R_3 - 20 \times 4 = 0 \implies 32 + R_3 - 80 = 0 \implies R_3 = 48\text{ kN} \end{aligned} $$ ## Determine maximum shear and maximum bending moment \[ \begin{aligned} &\text{Shear force } V(x) \text{ along the beam (from } A \text{ to } E\text{):}\ &\quad V(0^+) = R_1 = 58\text{ kN}\ &\quad V(2^-) = 58 - 20 \times 2 = 18\text{ kN}\ &\quad V(2^+) = 18 - 40 = -22\text{ kN}\ &\quad V(4^-) = -22\text{ kN}\ &\quad V(4^+) = -22 + R_2 = -22 - 10 = -32\text{ kN}\ &\quad V(5) = -32\text{ kN}\ &\quad V(6) = -32\text{ kN}\ &\quad V(10^-) = -32 - 20 \times 4 = -112\text{ kN}\ &\quad V(10^+) = -112 + R_3 = -112 + 48 = -64\text{ kN} \implies V_{\max} = 112\text{ kN}\ &\text{Bending moment } M(x) \text{ along the beam:}\ &\quad M(0) = 0\ &\quad M(2) = 58(2) - 20 \times 2 \times 1 = 76\text{ kN}\cdot\text{m}\ &\quad M(4) = 58(4) - 40(2) - 40(3) = 232 - 80 - 120 = 32\text{ kN}\cdot\text{m}\ &\quad M(5) = 0\text{ (at hinge } D\text{)}\ &\quad M(6) = -32(1) = -32\text{ kN}\cdot\text{m}\ &\quad M(10) = 0\ &\quad \text{Maximum positive moment: } M_{\max}^+ = 76\text{ kN}\cdot\text{m}\ &\quad \text{Maximum negative moment: } M_{\max}^- = -32\text{ kN}\cdot\text{m} \implies M_{\max} = 76\text{ kN}\cdot\text{m} \end{aligned} \] - **Reaction Forces:** - $R_1 = 58\text{ kN}$ (upward) - $R_2 = -10\text{ kN}$ (downward) - $R_3 = 48\text{ kN}$ (upward) - **Maximum Shear Force:** - $V_{\max} = 112\text{ kN}$ (at the left of support $R_3$) - **Maximum Bending Moment:** - $M_{\max} = 76\text{ kN}\cdot\text{m}$ (at $2\text{ m}$ from the left end $A$) { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Hinged Beam Analysis" ] }

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Analyze the beam segments and equilibrium of segment AC

Analyze segment DE and solve for reactions

Determine maximum shear and maximum bending moment

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