[{"data":1,"prerenderedAt":62},["ShallowReactive",2],{"q-113190-arch_structure-17":3},{"question":4,"neighbors":27,"sameNumber":46,"prev":35,"next":38},{"id":5,"classKey":6,"session":7,"year":8,"subject":9,"number":10,"groupStem":11,"ownStem":12,"stem":12,"options":13,"answer":18,"answerKind":20,"answerS":11,"answerNote":11,"images":21,"explanation":23,"withheld":11,"deepChars":24,"freq":25,"indexable":26},"arch-113190-801-0102-017","architect","113190",113,"建築結構",17,null,"圖示之二懸臂梁，短梁與長梁於 a 點之垂直變位量分別為 Δ1 與 Δ2，兩者關係可表示為 Δ1 = βΔ2，則 β 為何？ 2 EI 2L",{"A":14,"B":15,"C":16,"D":17},"1\u002F2","1\u002F4","1\u002F6","1\u002F8",[19],"B","single",[22],"\u002Fimages\u002Farch-113190-801-0102-017.png","本題考點：懸臂梁自由端受集中荷載時的端點變位公式 Δ = PL³\u002F(3EI)，以及變位與跨長三次方成正比、與撓曲剛度 EI 成反比的比例關係。\n【正解理由】依題圖，兩根梁都是 b 端固定、a 端自由，a 端各受向下集中荷載 P；短梁跨長 L、撓曲剛度 EI，長梁跨長 2L、撓曲剛度 2EI。懸臂梁自由端受集中荷載的變位為 Δ = PL³\u002F(3EI)。短梁 Δ1 = PL³\u002F(3EI)。長梁把跨長換成 2L、剛度換成 2EI：Δ2 = P(2L)³\u002F(3 × 2EI) = 8PL³\u002F(6EI) = 4PL³\u002F(3EI)。兩者相比，β = Δ1\u002FΔ2 = (1\u002F3)\u002F(4\u002F3) = 1\u002F4。跨長加倍使變位放大為 8 倍，剛度加倍又讓變位減半，淨效果是長梁變位為短梁的 4 倍，故選 B。\n【逐項排除】\n(A) 不符合。若誤用懸臂梁自由端轉角公式 θ = PL²\u002F(2EI) 去比較兩梁，得到的比值為 (1\u002F2)\u002F((2 × 2)\u002F(2 × 2)) = 1\u002F2，這是兩梁 a 端轉角之比，不是垂直變位之比。變位與跨長是三次方關係，不是平方關係，正確比值為 1\u002F4。\n(B) 符合。Δ1 = PL³\u002F(3EI)，Δ2 = 4PL³\u002F(3EI)，β = 1\u002F4。\n(C) 不符合。1\u002F6 與正確值 1\u002F4 不同，代表短梁變位被低估、或長梁變位被高估；依 Δ = PL³\u002F(3EI) 同時計入跨長三次方與剛度加倍兩個因素，比值只會是 1\u002F4。\n(D) 不符合。若只計入跨長加倍的三次方效應、漏掉長梁撓曲剛度為 2EI，會得 Δ2 = P(2L)³\u002F(3EI) = 8PL³\u002F(3EI)，比值 (1\u002F3)\u002F(8\u002F3) = 1\u002F8。長梁剛度加倍使變位減半，必須計入，正確比值為 1\u002F4。\n【演算步驟】\n公式：懸臂梁自由端受集中荷載 Δ = PL³\u002F(3EI)\n短梁：Δ1 = P·L³\u002F(3·EI)（單位 kN·m³\u002F(kN·m²) = m）\n長梁：Δ2 = P·(2L)³\u002F(3·2EI)，其中 (2L)³ = 2 × 2 × 2 × L³ = 8L³，3 × 2EI = 6EI，故 Δ2 = 8PL³\u002F(6EI) = (8\u002F6)·PL³\u002FEI = (4\u002F3)·PL³\u002FEI（單位 m）\n比值：β = Δ1\u002FΔ2 = (1\u002F3)\u002F(4\u002F3) = 0.25 = 1\u002F4\n【記憶點】懸臂端點變位 Δ = PL³\u002F(3EI)：跨長加倍變位乘 8，剛度加倍變位減半。",542,1,true,[28,31,34,37,40,43],{"number":29,"session":7,"stem":30},14,"一般建築結構耐震設計時，下列敘述何者錯誤？",{"number":32,"session":7,"stem":33},15,"有關建築結構耐震性質的敘述，下列何者錯誤？",{"number":35,"session":7,"stem":36},16,"一剛架結構受外力作用如圖所示，下列何種梁上塑性鉸產生時，不會形成不穩定結構？",{"number":38,"session":7,"stem":39},18,"格子梁系統中，兩直交梁的斷面相同且端點支承亦相同。若直向與橫向的跨距比為 1：2，則…",{"number":41,"session":7,"stem":42},19,"某懸臂梁長 3 m，承受均布載重 w。今將長度改為 6 m，仍承受均布載重 w，其他…",{"number":44,"session":7,"stem":45},20,"根據尤拉挫屈公式（Euler equation），下列何種作法可降低柱子的有效長度，…",[47,50,53,56,59],{"number":10,"session":48,"stem":49},"114180","圖中桁架結構有幾支零桿？",{"number":10,"session":51,"stem":52},"112190","關於鋼結構柱的挫屈，下列何者正確？",{"number":10,"session":54,"stem":55},"111180","關於在既有建築物的屋頂施做綠化工程前的考量，下列敘述何者正確？",{"number":10,"session":57,"stem":58},"110180","下列那棟建築物，較合乎永續性綠結構概念？",{"number":10,"session":60,"stem":61},"109180","在圖中的門形構架，設梁為剛體（EI = ∞），而質量 m 集中於剛體梁上。若柱的 E…",1791863197867]