This tool is licensed for use on the designated AEML, NTUST websites only.
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免責聲明
- 本工具由 AEML, NTUST 製作,專供「靜力學」、「材料力學」與「結構學」課程教學與自學使用。
- 計算採用線彈性、小變形假設之直接勁度法(梁理論),未考慮幾何與材料非線性、動力效應、應力集中及規範要求。
- 計算結果僅供學習參考,不得作為任何實際工程之分析、設計、施工或安全評估依據。
- 製作單位已盡力驗證程式之正確性,但不保證結果完全正確、完整或適用於特定目的;使用者應自行以其他方法驗算。
- 因使用或無法使用本工具所造成之任何直接或間接損失,製作單位概不負責。
- 本工具之程式碼與內容著作權屬 AEML, NTUST 所有,未經書面同意不得轉載、修改或作商業使用。
Disclaimer
- This tool is produced by AEML, NTUST for teaching and self-study in Statics, Mechanics of Materials and Structural Analysis.
- It uses the direct stiffness method with linear-elastic, small-displacement assumptions (beam theory). Geometric or material nonlinearity, dynamic effects, stress concentration and design-code requirements are not considered.
- Results are for learning only and must not be used for the analysis, design, construction or safety assessment of any real structure.
- Although the program has been verified with care, no warranty is given that the results are correct, complete or fit for any particular purpose. Users should check results independently.
- The authors accept no liability for any direct or indirect loss arising from the use of, or inability to use, this tool.
- The code and content are copyright of AEML, NTUST and may not be reproduced, modified or used commercially without written permission.
使用手冊(版本 1.4)
1. 快速開始
- 於左側工具盤點選元件,或把元件按鈕拖曳到工作區。
- 桿件:在工作區按住拖曳,或點起點再點終點。游標會吸附格點、既有節點與桿件。
- 支承與載重:點在節點或桿件上;支承點在桿件中段時,桿件會自動分段。
- 模型一有變動即自動重新計算,結果顯示在工作區與下方結果區。
- 上方「範例」選單內有各功能的典型例題,載入後會出現操作提示。
2. 工作區與選取
- 「選取」工具:點選物件以編輯屬性;拖曳節點或桿件可移動;拖曳空白處平移畫面。
- 縮放:滑鼠滾輪、雙指開合,或右上角 +、-、全覽。「最大化」隱藏輸入區與結果區;「全螢幕」適合手機。
- 多選:Shift+點選可加選桿件、節點與載重;「框選」工具或 Shift+拖曳空白處可框選。多選後可整組拖曳、刪除、套用材料與型鋼,載重可整批乘倍率。
- 複製、鏡射與陣列:屬性區可複製桿件(Ctrl+D)、左右或上下鏡射、依份數與間距建立陣列,載重與新節點上的支承一併複製。
- 載重可用滑鼠調整:拖曳箭頭移動位置,拖曳方塊控制點改變大小、方向或範圍;在分布載重上按右鍵可分段。
- 刪除:Delete 鍵、屬性區「刪除」或「刪除」工具。Ctrl+Z 復原、Ctrl+Y 重做、Esc 取消。
3. 元件
- 梁、剛架:兩端剛接,承受軸力、剪力與彎矩;可勾選「桿端鉸接」做出內鉸。桁架:兩端鉸接。
- 鉸支承束制 x、y;滾支承束制垂直支承面方向(可設傾斜角);固定端另束制轉角。
- 繩索只能受拉(鬆弛時以虛線顯示並失效);連桿為二力桿。光滑接觸面只提供指向結構的法向反力。滑套束制垂直導桿方向。彈簧提供 k 值彈性支承。
- 集中力:大小與方向角(由 +x 軸逆時針量,-90 度向下)。力矩:逆時針為正。均布與變均布載重:兩端強度、作用範圍與方向。
4. 輸入區
- 單位:SI (kN, m, MPa,斷面 mm)、公制 (kgw, cm)、英制 (lb, in, psi);切換時數值自動換算。
- 斷面:矩形、圓形、I 形、T 形、箱形、圓管、自訂 A 與 I、複合(底部貼板,以轉換斷面法計算)。「型鋼」選單有 H 型鋼、槽鋼、圓管與方管共 51 種。
- 材料:鋼、混凝土、鋁、木材會帶入 E、蒲松比、單位重、熱膨脹係數與降伏強度的典型值,皆可修改。G 由 E/[2(1+v)] 計算。
- 桿件另可輸入溫度變化、有效長度係數 K、降伏強度 fy,以及組合應力用的扭矩 T。
- 支承可輸入位移(沉陷),單位與位移相同,向下為負。
- 設定區:自重、剪力變形(Timoshenko)、應力顏色(藍拉、紅壓),以及載重組合(各載重組乘上係數後疊加,係數 0 可關閉該組)。
5. 結果與檢視
- 檢視列可開關載重、反力、變形(含動畫與倍率)、N/V/M 圖、編號、尺寸、格線與節點座標。
- 滑鼠移到桿件或內力圖上,即顯示該處的內力與上下緣應力。
- 結果區各區塊標題左側的三角形可收合或展開,狀態會記住。
- 表格:支承反力、節點位移、桿件內力與應力應變、挫屈臨界載重 Pcr 與安全係數(小於 1 以 !! 標示)。
6. 分析工具
- 自由體圖:未選取時為整體結構,選桿件或節點則為該自由體,並列出 ΣFx、ΣFy、ΣM 各項。
- 截面法:「截面法」工具拖一條切線,顯示箭頭所指一側的自由體與切斷處的 N、V、M。
- 斷面檢視:點選桿件任一位置,顯示內力、應力與應變分布;拖動「斷面高度」可看該點的應力元素、莫爾應力圓、莫爾應變圓與應變規讀數。
- 組合應力:桿件輸入扭矩 T 後,斷面檢視會把扭轉剪應力與彎曲剪應力合成,再求主應力。平面分析本身不含扭轉,T 視為沿桿件固定的外加扭矩。
- 內力圖:以課本式上下排列畫出 N、V、M。
- 影響線:選擇影響量後,單位載重沿路徑(以底色標示)移動;輸入輪重與輪距可求最大、最小效應並播放車列動畫。
- 勁度矩陣:顯示所選桿件的局部與整體元素矩陣;自由度不超過 18 時顯示整體 K 與位移向量。
- 塑性極限分析:勾選後列出各桿件的彈性模數 S、塑性模數 Z、形狀係數、My 與 Mp,並以逐步塑性鉸法求崩塌載重係數 λc 與塑性鉸形成順序(畫面上以紅色編號標示)。
7. 3D 空間桁架
- 設定區將「模型」改為 3D。可用桁架、繩索、連桿、鉸支承(球窩)、滾支承、節點力 (Fx, Fy, Fz) 與力偶。z 軸朝上,重力為 -z。
- 按住滑鼠在任何位置拖曳即可旋轉,Shift 或中鍵拖曳平移;設定區有等角、前視、俯視、側視按鈕。新節點落在可切換的工作平面上。
- 結果區列出 ΣF、ΣM 六條平衡式(選取節點則為節點法三條式子),並可計算力系對任意點與任意軸的力矩。
8. 計算器
- 形心與慣性矩:矩形與圓形組合(可挖空),求形心、Ix、Iy、Ixy 與主慣性矩,可套用為所選桿件的自訂斷面。
- 圓軸扭轉:階梯圓軸各段的內扭矩、最大剪應力與扭轉角。
- CSV 匯入匯出:以文字表格建立節點、桿件、支承與節點載重,適合大型桁架;視窗內會先列出目前模型作為格式範本。
9. 檔案與輸出
- 「存檔」下載 .json 模型檔,「讀檔」重新載入;瀏覽器會自動暫存最後的模型。
- 「分享連結」產生包含模型的網址。單位系統與原先不同時會提示。
- 工作區右上角「PNG」輸出圖檔;結果區「匯出 CSV」輸出結果表;「列印/PDF」產生含浮水印的報告版面。
10. 符號規定
- 桿件局部座標 x' 由起點指向終點(建立時取由左至右、由下至上),y' 為 x' 逆時針轉 90 度;「上緣」指 +y' 側。
- N 與應力以拉為正;M 以 -y' 側受拉(水平梁下緣受拉)為正;內力圖正值畫在 +y' 側。
11. 限制
- 線彈性、小變形;預設 Euler-Bernoulli 梁理論。挫屈僅提供分析平面內的 Euler 臨界載重。
- 塑性極限分析採比例加載、理想彈塑性、不計軸力與剪力對 Mp 的影響;分布載重下塑性鉸位置取八等分點,結果可能略高於理論值(約 2% 以內)。
- 矩形斷面的扭轉剪應力為近似值;自訂與複合斷面不支援扭轉與塑性模數。
- 3D 模式僅支援空間桁架;3D 力偶只計入力矩計算。
12. 版本紀錄
- 1.4(2026-11-13):結果區可收合、範例題庫(截面法、溫度、沉陷、挫屈、影響線、複合斷面、塑性、組合應力)、塑性模數與極限分析、扭矩與組合應力、CSV 匯入匯出、手冊重編。
- 1.3(2026-11-13):截面法、載重組合、溫度與沉陷、挫屈、勁度矩陣、莫爾應變圓、複合斷面、型鋼表擴充、多選節點與載重、車列動畫、3D 改為 z 軸朝上並新增力偶與力矩計算、形心與扭轉計算器。
- 1.2(2026-11-13):影響線、多選與框選、鏡射與陣列、材料庫、H 型鋼、懸停讀值、內力圖、列印/PDF、3D 空間桁架。
- 1.1(2026-11-12):自由體圖、莫爾圓、更多斷面、尺寸標註、桿件拖曳與複製、位移表、分享連結、CSV 匯出、剪力變形、載重拖曳與分段。
- 1.0(2026-11-12):2D 梁、桁架、剛架分析,三種語言與單位,存讀檔,全螢幕,動畫,PNG 輸出。
User Manual (version 1.4)
1. Quick start
- Pick a component in the left palette, or drag its button onto the workspace.
- Members: press and drag, or click the start point and then the end point. The cursor snaps to the grid, nodes and members.
- Supports and loads: click on a node or a member; a support placed inside a member splits it there.
- The model is re-analysed after every change; results appear in the workspace and the results panel.
- The Examples menu holds a typical problem for each feature, with a short tip after loading.
2. Workspace and selection
- Select tool: click an object to edit it; drag a node or member to move it; drag empty space to pan.
- Zoom with the wheel, a pinch, or the +, -, fit buttons. Maximise hides the side panels; Fullscreen suits phones.
- Multi-select: Shift+click adds members, nodes and loads; Box select or Shift+drag on empty space selects by box. A group can be dragged, deleted, given a material or steel shape, and its loads scaled.
- Copy, mirror and array: duplicate a member (Ctrl+D), mirror left-right or up-down, or repeat by count and spacing; loads and supports on new nodes follow.
- Loads can be adjusted with the mouse: drag the arrow to move it, drag the square handles to change size, direction or range; right-click a distributed load to split it.
- Delete with the Delete key, the Delete button or the Erase tool. Ctrl+Z undo, Ctrl+Y redo, Esc cancel.
3. Components
- Beam, Frame: rigid ends carrying axial force, shear and moment; tick "Hinge at end" for an internal hinge. Truss: pinned ends.
- Pin restrains x and y; Roller restrains the direction normal to its surface (inclination can be set); Fixed also restrains rotation.
- Cable is tension-only (dashed and inactive when slack); Link is a two-force member. Smooth surface gives only a normal push. Collar restrains motion normal to its rod. Spring gives stiffness k.
- Point load: magnitude and direction (counter-clockwise from +x; -90 degrees is downward). Moment: counter-clockwise positive. Distributed loads: end intensities, range and direction.
4. Input panel
- Units: SI (kN, m, MPa, section in mm), Metric (kgw, cm), Imperial (lb, in, psi); values are converted when switching.
- Sections: rectangular, circular, I, T, box, pipe, custom A and I, composite (bottom plate, transformed-section method). "Steel shape" lists 51 H-shapes, channels, pipes and tubes.
- Materials: steel, concrete, aluminium and timber fill in typical E, Poisson's ratio, unit weight, thermal coefficient and yield strength, all editable. G = E/[2(1+v)].
- Members also take a temperature change, effective length factor K, yield strength fy, and a torque T for combined stress.
- Supports take a prescribed displacement (settlement) in displacement units, downward negative.
- Settings: self-weight, shear deformation (Timoshenko), stress colour (blue tension, red compression), and load combination (cases times factors; a factor of 0 switches a case off).
5. Results and view
- The view bar toggles loads, reactions, deformed shape (animation and scale), N/V/M diagrams, labels, dimensions, grid and node coordinates.
- Hover over a member or a diagram to read internal forces and fibre stresses there.
- Each block in the results panel has a triangle at the left of its title to collapse or expand it; the state is remembered.
- Tables: reactions, node displacements, member forces, stresses and strains, Euler load Pcr and buckling factor (flagged with !! below 1).
6. Analysis tools
- Free-body diagram: the whole structure, or the selected member or joint, with every term of the ΣFx, ΣFy and ΣM equations.
- Method of sections: drag a line with the Section cut tool to see the free body on the side the arrows point to, with N, V, M at each cut.
- Section probe: click anywhere on a member for internal forces, stress and strain distributions; move "Height y" for the stress element, Mohr's circles of stress and strain, and rosette readings.
- Combined stress: with a torque T on the member, the probe combines torsional and bending shear before finding principal stresses. The plane analysis itself has no torsion; T is a constant applied torque.
- Stacked diagrams draw N, V and M textbook-style.
- Influence lines: choose a response; a unit load travels along the highlighted path, and axle loads with spacings give the extreme effects and a load-train animation.
- Stiffness matrices: local and global element matrices of the selected member, and the global K and displacement vector up to 18 degrees of freedom.
- Plastic limit analysis: lists elastic modulus S, plastic modulus Z, shape factor, My and Mp for each member, and finds the collapse load factor λc and the order of plastic hinges (numbered in red) by step-by-step hinge analysis.
7. 3D space truss
- Set Model to 3D. Truss, cable and link members, ball-and-socket and roller supports, nodal forces (Fx, Fy, Fz) and couples are available. The z axis is up; gravity is -z.
- Hold and drag anywhere to rotate, Shift or middle-drag to pan; Iso, Front, Top and Side buttons are in Settings. New nodes go on the selectable work plane.
- The results list the six equilibrium equations (three joint equations when a node is selected) and compute the moment of the force system about any point and axis.
8. Calculators
- Centroid and inertia: combine rectangles and circles (holes allowed) for the centroid, Ix, Iy, Ixy and principal values; the result can be applied to the selected member as a custom section.
- Shaft torsion: internal torque, maximum shear stress and twist of each segment of a stepped circular shaft.
- CSV import / export: build nodes, members, supports and nodal loads from a text table, handy for large trusses; the dialog shows the current model as a template.
9. Files and output
- Save downloads a .json model; Open loads it. The browser also keeps the last model.
- Share link creates a URL containing the model. A notice appears when its unit system differs.
- PNG exports the view; Export CSV saves the result tables; Print / PDF builds a watermarked report page.
10. Sign convention
- Local x' runs from start to end (left-to-right, bottom-to-top when created); y' is x' rotated 90 degrees counter-clockwise; "top" is the +y' side.
- N and stress are positive in tension; M is positive when the -y' side is in tension (sagging); positive values are plotted on the +y' side.
11. Limitations
- Linear elastic, small displacement; Euler-Bernoulli theory by default. Buckling is limited to the in-plane Euler load.
- Plastic limit analysis assumes proportional loading and elastic-perfectly-plastic behaviour, ignoring the effect of axial force and shear on Mp; under distributed load hinges form at eighth points, so the result can be slightly above theory (within about 2%).
- Torsional shear in rectangular sections is approximate; custom and composite sections do not support torsion or plastic modulus.
- 3D mode covers space trusses only; 3D couples enter the moment calculation only.
12. Version history
- 1.4 (2026-11-13): collapsible results, example bank (sections, temperature, settlement, buckling, influence line, composite, plastic, combined stress), plastic modulus and limit analysis, torque and combined stress, CSV import / export, manual reorganised.
- 1.3 (2026-11-13): method of sections, load combination, temperature and settlement, buckling, stiffness matrices, Mohr's circle of strain, composite section, more steel shapes, multi-select of nodes and loads, load-train animation, 3D z-up with couples and moment calculation, centroid and torsion calculators.
- 1.2 (2026-11-13): influence lines, multi-select, mirror and array, material library, H-shapes, hover read-out, stacked diagrams, Print / PDF, 3D space truss.
- 1.1 (2026-11-12): free-body diagram, Mohr's circle, more sections, dimensions, member drag and copy, displacement table, share link, CSV export, shear deformation, load dragging and splitting.
- 1.0 (2026-11-12): 2D beam, truss and frame analysis, three languages and unit systems, save / open, fullscreen, animation, PNG export.
應力模型:靜力學/材料力學/結構學|Stress Model: Statics / Mechanics of Materials / Structures
新檔|New
讀檔|Open
存檔|Save
復原|Undo
重做|Redo
範例…|Examples…
簡支梁|Simple beam
桁架|Truss
門型剛架|Portal frame
繩索懸吊梁|Cable-supported beam
中文|中文
中文/English|中文/English
English|English
全螢幕|Fullscreen
手冊|Manual
免責聲明|Disclaimer
PNG|PNG
我已閱讀並同意|I have read and agree
關閉|Close
編輯|Edit
桿件|Members
支承|Supports
載重|Loads
選取|Select
刪除|Erase
梁|Beam
桁架|Truss
剛架|Frame
鉸支承|Pin
滾支承|Roller
固定端|Fixed
繩索|Cable
連桿|Link
光滑面|Smooth
滑套|Collar
彈簧|Spring
集中力|Point load
均布載重|UDL
變均布載重|Varying load
力矩|Moment
設定|Settings
模型|Model
2D|2D
3D 空間桁架|3D space truss
單位|Units
SI (kN, m, MPa)|SI (kN, m, MPa)
公制 (kgw, cm)|Metric (kgw, cm)
英制 (lb, in, psi)|Imperial (lb, in, psi)
格點間距|Grid step
吸附格點|Snap to grid
自重|Self-weight
應力顏色|Stress colour
屬性輸入|Properties
以「選取」工具點選桿件、節點、支承或載重,即可在此輸入數值。|Use the Select tool to click a member, node, support or load and edit its values here.
節點|Node
x|x
y|y
桿件|Member
長度|Length
型式|Type
斷面積 A|Area A
斷面型式|Section
矩形|Rectangular
圓形|Circular
寬 b|Width b
深 h|Depth h
直徑 d|Diameter d
A|A
I|I
G(自動)|G (auto)
E|E
蒲松比 ν|Poisson ν
單位重 γ|Unit weight γ
起點鉸接|Hinge at start
終點鉸接|Hinge at end
傾斜角|Inclination
導桿角度|Rod angle
彈簧常數 k|Stiffness k
束制轉角|Restrain rotation
束制 x、y 位移。|Restrains x and y translation.
束制垂直於支承面的位移;傾斜角以逆時針為正。|Restrains motion normal to the surface; inclination is counter-clockwise positive.
束制位移與轉角。|Restrains translation and rotation.
僅能提供指向結構的法向反力,脫離時失效。|Normal reaction pushing on the structure only; released on lift-off.
束制垂直於導桿方向的位移。|Restrains motion normal to the rod.
沿支承法線方向的彈性支承。|Elastic support along the support normal.
大小 P|Magnitude P
方向角|Direction
力矩 M(逆時針為正)|Moment M (CCW +)
起點強度 w1|Start w1
終點強度 w2|End w2
變均布|Varying
起點位置 (0~1)|Start (0-1)
終點位置 (0~1)|End (0-1)
方向|Direction
重力方向|Gravity
水平 +x|Horizontal +x
垂直桿件|Normal to member
位置 (0~1)|Position (0-1)
刪除|Delete
載重|Loads
反力|Reactions
變形|Deformed
動畫|Animation
速度|Speed
自動|Auto
慢|Slow
中|Medium
快|Fast
變形倍率|Scale
自動|Auto
N 圖|N
V 圖|V
M 圖|M
編號|Labels
結果|Results
結構不穩定(支承不足或為機構),無法求解|Unstable structure (insufficient supports or a mechanism)
請先建立桿件。|Add members to begin.
穩定|Stable
靜定|Statically determinate
靜不定|Statically indeterminate
最大位移|Max displacement
斷面檢視|Section probe
項目|Item
數值|Value
位置 x′|Position x′
軸向應力 N/A|Axial stress N/A
上緣應力 σ|Top stress σ
下緣應力 σ|Bottom stress σ
最大剪應力 τ|Max shear stress τ
上緣應變 ε|Top strain ε
下緣應變 ε|Bottom strain ε
最大剪應變 γ|Max shear strain γ
側向應變 -νε(軸向)|Lateral strain -νε (axial)
以「選取」工具點選桿件上任一位置,可檢視該斷面的內力、應力與應變。|With the Select tool, click any point on a member to inspect internal forces, stress and strain there.
支承反力|Support reactions
桿件內力、應力、應變與變形|Member forces, stress, strain and deformation
彈簧變形/狀態|Spring def. / state
未作用|inactive
N 拉為正;σ 拉為正;ε 為最大正向應力對應之應變;δ 為實際位移(未放大)。|N and σ positive in tension; ε is the strain at the largest normal stress; δ is the true (unscaled) displacement.
■ 拉應力|■ Tension
■ 壓應力|■ Compression
請點在節點或桿件上。|Click on a node or a member.
請點在桿件上。|Click on a member.
請點選終點(Esc 取消)。|Click the end point (Esc to cancel).
檔案格式不符。|Not a valid model file.
尺寸|Dimensions
格線|Grid
剪力變形|Shear deformation (Timoshenko)
I 形|I-section
T 形|T-section
箱形|Box
圓管|Pipe
自訂 A、I|Custom A, I
腹板厚 tw|Web tw
翼板厚 tf|Flange tf
壁厚 t|Wall t
斷面積 A|Area A
慣性矩 I|Inertia I
複製桿件|Duplicate (Ctrl+D)
於中點分段|Split at midpoint
拖曳箭頭可沿桿件或節點移動位置;拖曳尾端方塊可調整大小與方向。|Drag the arrow to move it along members or to a node; drag the square handle to change magnitude and direction.
拖曳載重:沿桿件方向為移動位置,垂直方向為調整大小;拖曳兩端方塊可調整範圍與端點強度;在載重上按右鍵可於該處分段,各段可分別設定大小。|Drag the load along the member to move it or across to resize it; drag the end handles to change the range and end intensity; right-click on the load to split it there so each part can be set separately.
已分段,各段可分別選取並設定大小。|Split done; select each part to set its size.
此位置無法分段。|Cannot split at this position.
分享連結|Share link
此連結已包含目前的模型,開啟後即載入。模型較大時連結會較長。|This link contains the current model and loads it when opened. Large models give long links.
複製連結|Copy link
已複製|Copied
匯出 CSV|Export CSV
自由體圖與平衡方程式|Free-body diagram and equilibrium
整體結構|Whole structure
桿件|Member
節點|Joint
選取桿件或節點可切換為該自由體;未選取時為整體結構。紅色為外力、綠色為支承反力、紫色為桿端內力;分布載重以等效合力表示,力矩以黑點為中心、逆時針為正。|Select a member or node to isolate it; with nothing selected the whole structure is shown. Red: applied loads, green: reactions, violet: member end forces. Distributed loads are shown as resultants; moments are taken about the black dot, counter-clockwise positive.
斷面高度 y|Height y
該點正向應力 σ|Normal stress σ at y
該點剪應力 τ|Shear stress τ at y
最大面內剪應力|Max in-plane shear
應力元素與莫爾圓|Stress element and Mohr's circle
節點位移|Node displacements
框選|Box select
列印/PDF|Print / PDF
3D 三腳架桁架|3D tripod truss
3D 三繩懸吊(空間力系)|3D three-cable hanger
此模型的單位系統為:|This model uses the unit system:
材料|Material
自訂|Custom
鋼|Steel
混凝土|Concrete
鋁|Aluminium
木材|Timber
型鋼 (mm)|Steel shape (mm)
選擇規格…|Pick a size…
複製、鏡射與陣列|Copy, mirror and array
左右鏡射|Mirror left-right
上下鏡射|Mirror up-down
陣列份數|Array copies
間距 dx|Spacing dx
間距 dy|Spacing dy
建立陣列|Create array
已選取桿件數|Members selected
(不變更)|(keep)
沒有可建立的桿件(位置重複)。|Nothing created (members already exist there).
Shift+點選可加選桿件;「框選」工具或 Shift+拖曳空白處可框選。拖曳任一已選桿件可整組移動。鏡射以選取範圍的右緣或下緣為軸。|Shift+click adds members; use Box select or Shift+drag on empty space to select by box. Drag any selected member to move the group. Mirroring uses the right or bottom edge of the selection as the axis.
內力圖|N / V / M diagrams
滑鼠移到圖上可讀取該位置的數值。|Move the pointer over the diagrams to read values.
影響線與移動載重|Influence line and moving loads
(不顯示)|(off)
影響量|Response
輪重 P1, P2…|Axle loads P1, P2…
輪距 s1, s2…|Axle spacings s1, s2…
車列|Load train
單位載重(向下 1)沿路徑移動,影響線以虛線畫在工作區。路徑預設為所有梁與剛架的非垂直桿件(無則取最低的水平桿件);多選桿件可自訂路徑。內力與位移的影響線需先點選桿件位置。輪重與輪距以逗號分隔,x1 為第一個輪的位置。含繩索或光滑面時結果僅供參考。|A unit load (1, downward) travels along the path; the influence line is drawn dashed in the workspace. The default path is every non-vertical beam or frame member (otherwise the lowest horizontal members); multi-select members to define your own path. Influence lines for internal forces or deflection need a probed section. Separate axle loads and spacings with commas; x1 is the position of the first axle. With cables or smooth surfaces the result is indicative only.
列印或存成 PDF|Print or save as PDF
在列印對話框選擇「另存為 PDF」即可輸出 PDF。|Choose "Save as PDF" in the print dialog to export a PDF.
僅供教學使用,計算結果不得作為工程設計依據。 For education only; not for engineering design.|For education only; not for engineering design.
工作平面|Work plane
XZ 平面 (y)|XZ plane (y)
XY 平面 (z)|XY plane (z)
YZ 平面 (x)|YZ plane (x)
平面位置 y|Plane position y
平面位置 z|Plane position z
平面位置 x|Plane position x
3D 模式:按住滑鼠拖曳即可旋轉視角(任何位置皆可),Shift 或中鍵拖曳平移,滾輪縮放;其他工具下可用右鍵拖曳旋轉。新節點落在工作平面上,或吸附到既有節點;座標可在此直接修改。z 軸朝上,重力為 -z。|3D mode: hold and drag anywhere to rotate the view, Shift or middle-button drag to pan, wheel to zoom; with other tools, right-drag rotates. New nodes are placed on the work plane or snap to existing nodes; coordinates can be edited here. The z axis is up and gravity acts along -z.
z|z
束制 x|Restrain x
束制 y|Restrain y
束制 z|Restrain z
球窩支承束制 x、y、z;取消勾選可成為滾支承或滑槽。|A ball-and-socket restrains x, y and z; untick directions to make a roller or slot.
合力大小|Magnitude
方向餘弦 l, m, n|Direction cosines l, m, n
已切換為 3D 空間桁架模式。|Switched to 3D space truss mode.
已切換為 3D;桿件上的載重與力矩在 3D 不支援,已移除(可復原)。|Switched to 3D; member loads and moments are not supported in 3D and were removed (undo to restore).
已切換為 2D;z 座標已歸零。|Switched to 2D; z coordinates were set to zero.
請點在節點上。|Click on a node.
目前視角與工作平面平行,請旋轉視角或更換工作平面。|The view is edge-on to the work plane; rotate the view or change the plane.
空間力系平衡方程式|Equilibrium of the space force system
各項依序為相連桿件的軸力分量、節點外力、支承反力。|Terms are member force components, applied load and reaction, in that order.
各項依序為外力(含自重)與支承反力;力矩取自|Terms are applied loads (with self-weight) then reactions; moments are taken about
空間結構不穩定(支承或桿件不足,無法求解)|Unstable space structure (not enough supports or members)
桿件軸力、應力、應變與伸長量|Member axial force, stress, strain and elongation
狀態|State
零力桿|Zero-force
拉 T|Tension
壓 C|Compression
l, m, n 為桿件方向餘弦(由起點指向終點);N 與 σ 以拉為正;3D 模式僅分析軸力桿件。|l, m, n are direction cosines from start to end; N and σ are positive in tension; only axial members are analysed in 3D.
截面法|Section cut
清除切面|Clear section cut
截面法(切面箭頭所指一側)|Method of sections (side the arrows point to)
切點|cut point
取此側|this side
座標|Coordinates
拖曳可旋轉|drag to rotate
載重組合|Load combination
新載重歸屬|New loads go to
新增載重組|Add load case
載重組|Load case
每個載重屬於一個載重組 LC;分析時各組乘上其係數後疊加。係數設為 0 可暫時關閉該組,用來示範疊加原理。自重不受係數影響。|Each load belongs to a load case LC; the analysis superposes the cases times their factors. Set a factor to 0 to switch a case off and demonstrate superposition. Self-weight is not factored.
熱膨脹係數 α|Thermal coeff. α
溫度變化 ΔT|Temperature change ΔT
有效長度係數 K|Effective length factor K
支承位移 δx|Settlement δx
支承位移 δy|Settlement δy
支承位移 δz|Settlement δz
挫屈安全係數 Pcr/N|Buckling factor Pcr/N
顯示勁度矩陣|Show stiffness matrices
單位:|Units:
點選桿件可顯示其元素勁度矩陣。|Select a member to see its element stiffness matrix.
自由度超過 18,不顯示整體勁度矩陣。|More than 18 degrees of freedom; the global matrix is not shown.
元素勁度矩陣(局部座標,已含桿端鉸接修正)|Element stiffness (local axes, end hinges condensed)
元素勁度矩陣(整體座標)|Element stiffness (global axes)
整體勁度矩陣 K(未含支承束制與彈簧)|Global stiffness K (before supports and springs)
莫爾應變圓|Mohr's circle of strain
該點 εx|εx at point
該點 εy = -νεx|εy = -νεx at point
該點 γxy|γxy at point
最大面內剪應變 γmax|Max in-plane shear strain γmax
應變規 0°/45°/90°|Rosette 0°/45°/90°
複合斷面(底部貼板)|Composite (bottom plate)
底板寬 b2|Plate width b2
底板厚 h2|Plate thickness h2
底板 E2|Plate E2
已選取節點數|Nodes selected
已選取載重數|Loads selected
載重倍率|Load multiplier
套用倍率到所選載重|Scale selected loads
車列動畫|Animate load train
力偶以雙箭頭表示(右手定則)。空間桁架無法承受力偶,此力偶只計入下方的力矩計算,不參與桁架分析。|A couple is drawn as a double-headed arrow (right-hand rule). A space truss cannot resist couples, so it enters only the moment calculation below, not the truss analysis.
對任意點與任意軸之力矩|Moment about a point and an axis
軸向|Axis
計入支承反力|Include reactions
平行於合力之力矩分量|Moment component parallel to R
輸入點 O 的座標與軸的方向向量 (a, b, c)。R 為外力合力,M_O 為外力與力偶對 O 的合力矩,M_axis 為其在該軸上的分量。|Enter point O and the axis direction (a, b, c). R is the resultant of the applied forces, M_O the resultant moment of forces and couples about O, and M_axis its component along the axis.
等角|Iso
前視|Front
俯視|Top
側視|Side
計算器…|Calculators…
形心與慣性矩計算器|Centroid and inertia calculator
圓軸扭轉計算器|Shaft torsion calculator
形狀|Shape
寬 b / 直徑 d|Width b / dia. d
高 h|Height h
形心 x|Centroid x
形心 y|Centroid y
加/挖空|Add / hole
新增一塊|Add part
輸入各矩形或圓形的尺寸與其形心座標;「−」表示挖空。長度單位|Enter each rectangle or circle with the coordinates of its own centroid; "−" makes a hole. Length unit
套用為所選桿件的自訂斷面 (A, Ix, 深度, 形心位置)|Apply to the selected member as a custom section (A, Ix, depth, centroid)
先在工作區選取一根桿件,即可把計算結果套用為該桿件的自訂斷面。|Select a member in the workspace first to apply the result as its custom section.
已套用為自訂斷面。|Applied as a custom section.
長度 L|Length L
外徑 do|Outer dia. do
內徑 di|Inner dia. di
段右端外加扭矩 T|Torque T at right end
新增一段|Add segment
階梯圓軸左端固定,由左至右分段;各段右端可施加扭矩。di = 0 為實心軸。τ = Tc/J,φ = TL/(GJ)。|A stepped circular shaft fixed at its left end, segments listed left to right; a torque may act at the right end of each segment. di = 0 is a solid shaft. τ = Tc/J, φ = TL/(GJ).
內扭矩 T|Internal torque T
最大剪應力 τmax|Max shear stress τmax
段扭轉角 φ|Segment twist φ
累計扭轉角|Total twist
截面法:桁架|Method of sections: truss
溫度應力:兩端固定桿|Thermal stress: fixed-fixed bar
支承沉陷:兩跨連續梁|Settlement: two-span beam
挫屈:二桿桁架|Buckling: two-bar truss
影響線:兩跨連續梁|Influence line: two-span beam
複合斷面:木梁貼鋼板|Composite: timber with steel plate
塑性極限:兩端固定梁|Plastic limit: fixed-fixed beam
組合應力:受彎扭圓軸|Combined stress: shaft in bending and torsion
切線已畫好:結果區的自由體圖顯示左半部與三根被切斷桿件的軸力。用「截面法」工具可重畫切線。|A cut is already drawn: the free body shows the left part and the three cut member forces. Redraw it with the Section cut tool.
桿件升溫 40°C、兩端固定:軸力為 -EAαΔT。把一端改成滾支承,軸力即消失。|The bar is heated by 40°C between fixed ends: N = -EAαΔT. Change one end to a roller and the force disappears.
中間支承下沉 10 mm、沒有外力:靜不定結構仍產生彎矩與反力。點選中間支承可修改位移。|The middle support settles 10 mm with no load: the indeterminate beam still develops moments and reactions. Select the middle support to change it.
水平桿受壓 16 kN,桿件表的挫屈安全係數小於 1(標示 !!)。把直徑加大或改變 K 值觀察變化。|The horizontal strut carries 16 kN in compression and its buckling factor is below 1 (flagged !!). Increase the diameter or change K to see the effect.
已顯示中間支承反力的影響線與三軸車列的最大效應位置。在結果區「影響線」可更換影響量或勾選車列動畫。|The influence line of the middle reaction and the worst position of a three-axle train are shown. Change the response or tick the animation in the Influence line block.
木梁底部貼鋼板:斷面檢視的應力圖在交界處跳躍,鋼板應力為轉換斷面應力乘上模數比。|Timber beam with a steel plate underneath: the stress diagram jumps at the interface; plate stress is the transformed stress times the modular ratio.
結果區「塑性極限分析」列出 Z、形狀係數、Mp 與崩塌載重係數 λc;紅色編號為塑性鉸形成順序(理論值 wc = 16Mp/L²)。|The Plastic limit analysis block lists Z, shape factor, Mp and the collapse factor λc; red numbers give the hinge order (theory: wc = 16Mp/L²).
懸臂圓軸同時受彎矩與扭矩 T:拖動「斷面高度」比較上緣(彎曲應力最大)與中性軸(剪應力最大)的主應力。|A cantilever shaft under bending and a torque T: move "Height y" to compare principal stresses at the top fibre and at the neutral axis.
塑性極限分析|Plastic limit analysis
沒有可形成塑性鉸的梁或剛架桿件(需有塑性模數與 fy)。|No beam or frame member can form a plastic hinge (plastic modulus and fy are needed).
形狀係數 Z/S|Shape factor Z/S
目前載重型式不會形成崩塌機構。|No collapse mechanism forms under this load pattern.
崩塌載重 = 目前載重 × λc|collapse load = current loads × λc
塑性鉸順序|Hinge order
形成時的載重係數 λ|Load factor λ at formation
所有載重(含自重)依同一比例放大;塑性鉸依序形成,直到結構成為機構。不計軸力與剪力對 Mp 的影響;分布載重下塑性鉸位置取八等分點。|All loads (self-weight included) are scaled together; hinges form one after another until a mechanism results. Axial force and shear effects on Mp are ignored; under distributed load hinges form at eighth points.
降伏強度 fy|Yield strength fy
扭矩 T(組合應力用)|Torque T (combined stress)
彎曲剪應力 τV|Bending shear τV
扭轉剪應力 τT|Torsional shear τT
扭轉常數 J|Torsion constant J
全桿扭轉角|Twist over the member
此斷面不支援扭轉計算|Torsion is not supported for this section
CSV 匯入/匯出模型|CSV model import / export
載入 CSV 檔|Load a CSV file
匯入(取代目前模型)|Import (replaces the model)
下載 CSV|Download CSV
CSV 內容無法辨識:需要 NODES 與 MEMBERS 兩個區段。|Cannot read the CSV: NODES and MEMBERS sections are required.
已匯入節點/桿件:|Imported nodes / members:
下方為目前模型,可直接修改或貼上自己的表格。區段標題為 NODES、MEMBERS、SUPPORTS、LOADS;座標與力使用目前的單位系統。桿件型式可為 beam、truss、frame、cable、link(省略時為 truss);支承型式為 pin、roller、fixed。任一節點的 z 不為 0 時會建立為 3D 模型。匯入可用「復原」取消。|The current model is shown below; edit it or paste your own table. Section titles are NODES, MEMBERS, SUPPORTS and LOADS; coordinates and forces use the current unit system. Member type may be beam, truss, frame, cable or link (truss if omitted); support type is pin, roller or fixed. If any node has a non-zero z the model becomes 3D. Import can be undone.