This tool is licensed for use on the designated AEML, NTUST websites only.
Integrity check failed. Please use the original version on the official AEML, NTUST page.
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),未考慮挫屈、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). Buckling, 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
1. 快速開始
- 於左側工具盤點選元件(或直接把元件按鈕拖曳到工作區)。
- 桿件:在工作區按住拖曳,或點一下起點、再點一下終點。游標會吸附到格點、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,桿件會自動在該處分段。
- 模型一有變動即自動重新計算,結果直接顯示在工作區與下方結果區。
- 上方「範例」選單可載入簡支梁、桁架、剛架、懸吊梁等範例。
2. 工作區操作
- 「選取」工具:點選物件以編輯屬性;拖曳節點可移動;拖曳空白處可平移畫面。
- 縮放:滑鼠滾輪、雙指開合,或右上角的 +、-、全覽按鈕。
- 刪除:選取後按 Delete 鍵或屬性區的「刪除」,也可使用「刪除」工具。Ctrl+Z 復原、Ctrl+Y 重做、Esc 取消。
- 右上角「最大化」可隱藏輸入區與結果區;上方「全螢幕」適合手機檢視。
3. 元件說明
- 梁、剛架:兩端剛接,可承受軸力、剪力與彎矩;可在屬性區勾選「桿端鉸接」做出內鉸。
- 桁架:兩端鉸接,主要承受軸力。
- 鉸支承:束制 x、y 位移。滾支承:束制垂直於支承面方向的位移,可設定傾斜角。固定端:束制位移與轉角。
- 繩索:只能受拉,受壓時自動失效(以虛線顯示)。連桿:可受拉或受壓的二力桿。兩者需輸入 E 與斷面積,拖曳終點落在空白處時會自動加上鉸接錨定點。
- 光滑接觸面:只能提供指向結構的法向反力,脫離時自動失效。
- 滑套:束制垂直於導桿方向的位移,可選擇是否同時束制轉角。
- 彈簧:沿設定方向提供 k 值的彈性支承。
- 集中力:輸入大小與方向角(由 +x 軸逆時針量,-90 度為向下)。力矩:逆時針為正。
- 均布與變均布載重:輸入起點與終點強度及作用範圍(0 到 1 為桿長比例);方向可選重力方向、水平方向或垂直桿件。
4. 輸入區
- 單位系統:SI (kN, m, MPa,斷面尺寸 mm)、公制 (kgw, cm, kgw/cm2)、英制 (lb, in, psi)。切換時既有數值會自動換算。
- 桿件:斷面可選矩形(寬 b、深 h,h 為彎曲方向)或圓形(直徑 d);輸入 E 與蒲松比,G 由 E/[2(1+v)] 自動計算。
- 自重:開啟後可輸入各桿件單位重,以沿桿長的均布載重計入。
- 應力顏色:開啟後桿件以顏色顯示正向應力,藍色為拉、紅色為壓;梁與剛架的上下緣分別著色。
5. 結果區與符號規定
- 可切換顯示載重、支承反力、變形、軸力圖 N、剪力圖 V、彎矩圖 M 與編號。
- 變形以放大倍率繪製,可自動或手動設定;動畫速度可調整。
- 點選桿件任一位置(或拖動滑桿),顯示該斷面的內力、應力、應變與位移,並繪出斷面正向應力與剪應力分布。
- 桿件局部座標:x' 由起點指向終點(建立時自動取由左至右、由下至上),y' 為 x' 逆時針轉 90 度。「上緣」指 +y' 側。
- N 以拉力為正;M 以使 -y' 側受拉(水平梁下緣受拉)為正;內力圖正值畫在 +y' 側。應力以拉為正。
- 結果區同時顯示結構為靜定或靜不定(括號內為靜不定次數);結構不穩定時會提示。
6. 存檔、讀檔與輸出
- 「存檔」下載 .json 模型檔,「讀檔」可重新載入;瀏覽器也會自動暫存最後的模型。
- 工作區右上角「PNG」可將目前畫面輸出為圖檔。
7. 限制
- 目前為 2D 平面分析,3D 功能將於後續版本提供。
- 採線彈性、小變形假設;預設為 Euler-Bernoulli 梁理論(可選擇計入剪力變形),不含挫屈檢核。
8. 進階功能
- 斷面:除矩形、圓形外,可選 I 形、T 形(翼板在 +y' 側)、箱形、圓管,或直接輸入 A、I 與深度。
- 剪力變形:於設定勾選後,以 Timoshenko 梁理論計入剪力變形(剪力面積由斷面形狀係數計算,自訂斷面取 1.2)。
- 自由體圖:結果區會畫出整體結構、所選桿件或所選節點的自由體圖,並列出 ΣFx、ΣFy、ΣM 的各項數值。
- 應力元素與莫爾圓:在斷面檢視中拖動「斷面高度 y」滑桿,可查看該點的 σ、τ、主應力、最大面內剪應力與主平面角。
- 滑鼠調整載重:以「選取」工具點選載重後,可直接拖曳移動;拖曳方塊控制點可改變大小、方向或作用範圍。
- 分布載重分段:在分布載重上按右鍵(或按屬性區的「於中點分段」),可把載重切成兩段,各段分別設定大小。
- 桿件拖曳與複製:以「選取」工具拖曳桿件可整體移動;屬性區的「複製桿件」或 Ctrl+D 可複製桿件及其載重。
- 尺寸與格線:檢視列可開關尺寸標註與格線,輸出 PNG 時同樣適用。
- 分享連結:產生包含目前模型的網址,對方開啟即可看到相同模型。
- 匯出 CSV:將反力、節點位移與桿件結果匯出為試算表可讀的檔案。
User Manual
1. Quick start
- Pick a component in the left palette (or drag its button onto the workspace).
- Members: press and drag in the workspace, or click the start point and then the end point. The cursor snaps to the grid, existing nodes and members.
- Supports and loads: click on a node or on a member. A support placed inside a member splits the member there.
- The model is re-analysed automatically after every change; results appear in the workspace and in the results panel.
- The Examples menu loads a simple beam, a truss, a frame and a cable-supported beam.
2. Workspace
- Select tool: click an object to edit it; drag a node to move it; drag empty space to pan.
- Zoom with the mouse wheel, a two-finger pinch, or the +, -, fit buttons.
- Delete with the Delete key, the Delete button, or the Erase tool. Ctrl+Z undo, Ctrl+Y redo, Esc cancel.
- Maximise hides the input and results panels; Fullscreen is convenient on phones.
3. Components
- Beam, Frame: rigid ends carrying axial force, shear and moment. Tick "Hinge at end" to create an internal hinge.
- Truss: pinned at both ends, mainly axial force.
- Pin: restrains x and y. Roller: restrains the direction normal to its surface (inclination can be set). Fixed: restrains translation and rotation.
- Cable: tension only; it is removed automatically when slack (dashed line). Link: two-force member in tension or compression. Both need E and area. If the far end is dropped on empty space a pinned anchor is added.
- Smooth surface: provides only a normal reaction pushing on the structure; it is released on lift-off.
- Collar on smooth rod: restrains motion normal to the rod; rotation may also be restrained.
- Spring: elastic support of stiffness k along the given direction.
- Point load: magnitude and direction angle (counter-clockwise from +x; -90 degrees is downward). Moment: counter-clockwise positive.
- Uniform and varying distributed loads: start/end intensity and range (0 to 1 as a fraction of member length); direction may be gravity, horizontal, or normal to the member.
4. Input panel
- Units: SI (kN, m, MPa, section in mm), Metric (kgw, cm, kgw/cm2), Imperial (lb, in, psi). Existing values are converted when switching.
- Members: rectangular (width b, depth h in the bending direction) or circular (diameter d) section; enter E and Poisson's ratio, G = E/[2(1+v)] is computed.
- Self-weight: when on, enter the unit weight of each member; it is applied as a distributed load.
- Stress colour: members are coloured by normal stress, blue for tension and red for compression; top and bottom fibres of beams are coloured separately.
5. Results and sign convention
- Toggle loads, reactions, deformed shape, N, V, M diagrams and labels.
- The deformed shape is magnified by an automatic or manual factor; animation speed is adjustable.
- Click anywhere on a member (or move the slider) to see internal forces, stresses, strains and displacement at that section, with normal and shear stress distributions.
- Local axes: x' runs from start to end (left-to-right, bottom-to-top when created); y' is x' rotated 90 degrees counter-clockwise. "Top" means the +y' side.
- N is positive in tension; M is positive when the -y' side is in tension (sagging for a horizontal beam); positive values are plotted on the +y' side. Stress is positive in tension.
- The panel reports whether the structure is statically determinate or indeterminate (degree in brackets) and warns when it is unstable.
6. Save, open and export
- Save downloads a .json model file; Open loads it again. The browser also keeps the last model automatically.
- PNG exports the current view as an image.
7. Limitations
- 2D plane analysis only; 3D will be added in a later version.
- Linear elastic, small displacement; Euler-Bernoulli theory by default (shear deformation optional); no buckling check.
8. Advanced features
- Sections: besides rectangular and circular, choose I, T (flange on the +y' side), box, pipe, or enter A, I and depth directly.
- Shear deformation: tick it in Settings to use Timoshenko beam theory (shear area from the section form factor; 1.2 is used for custom sections).
- Free-body diagram: the results panel draws the whole structure, the selected member or the selected joint, and lists every term of the ΣFx, ΣFy and ΣM equations.
- Stress element and Mohr's circle: move the "Height y" slider in the section probe to see σ, τ, principal stresses, maximum in-plane shear and the principal angle at that point.
- Adjusting loads with the mouse: select a load with the Select tool and drag it to move it; drag its square handles to change magnitude, direction or range.
- Splitting a distributed load: right-click on it (or use "Split at midpoint") to cut it into two parts that can be sized separately.
- Moving and duplicating members: drag a member with the Select tool to move it; "Duplicate member" or Ctrl+D copies it with its loads.
- Dimensions and grid can be toggled in the view bar; the PNG export follows the same settings.
- Share link: creates a URL containing the current model.
- Export CSV: saves reactions, node displacements and member results for a spreadsheet.
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(後續版本)|3D (later version)
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
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