Analysis and Post-Processing - Theory & Concepts

Analysis and Post-Processing

The core purpose of building a model and applying loads is to determine how the structure responds. This module explains the mathematical processes STAAD Pro uses to calculate unknown displacements and internal forces, and subsequently visualizing that data before proceeding to actual member design.

Executing the Analysis

The true power of STAAD Pro lies in its highly advanced analysis engine. This engine primarily utilizes the generalized Matrix Stiffness Method, fundamentally relating applied forces to resulting displacements through a massive, complex system of simultaneous linear equations.

Perform Analysis

The standard command (PERFORM ANALYSIS) in the STAAD Editor that officially instructs the analysis engine to construct the global stiffness matrix from all member stiffnesses, solve the matrix for unknown joint displacements (using techniques like Cholesky decomposition), and subsequently back-calculate the resulting internal member forces and support reactions based on those displacements.

The Matrix Stiffness Method

The entire static linear elastic analysis hinges on solving the fundamental equilibrium equation of the entire assembled structure:

The Matrix Stiffness Equation

The governing equation of linear structural analysis, solved for the unknown displacements \{D\}.

[K]\{D\} = \{F\}

Variables

Symbol	Description	Unit
$[K]$	The Global Stiffness Matrix (square, symmetric, and positive definite if properly supported)	-
$\{D\}$	The Global Displacement Vector (unknown nodal translations and rotations)	-
$\{F\}$	The Global Force Vector (known applied loads and unknown support reactions)	-

The Solution Process

Element Stiffness Formulation: STAAD calculates the local stiffness matrix $[k]$ for every individual element based on its properties ( $A, E, I, L, G, J$ ).
Coordinate Transformation: Each local matrix is transformed into the global coordinate system using a direction cosine transformation matrix $[T]$ such that $[k]_{global} = [T]^T [k]_{local} [T]$ .
Global Assembly: The individual global matrices are mathematically superimposed (added together at common nodes) to form the massive Global Stiffness Matrix $[K]$ . This matrix represents the total stiffness of the entire interconnected frame.
Boundary Condition Application: The matrix $[K]$ is initially singular (cannot be solved because the structure can "fly away" as a rigid body). STAAD applies the user-defined support conditions (e.g., setting specific displacements in $\{D\}$ to exactly zero). This removes rows and columns, partitioning the matrix into a non-singular format.
Solving for Displacements: The partitioned matrix is inverted (or solved via Gaussian elimination/Cholesky factorization) to find the unknown joint displacements: $\{D\}_{unknown} = [K]_{partition}^{-1} \{F\}_{applied}$ .
Back-Substitution: The newly found displacements are multiplied back against each individual element's stiffness matrix to determine the internal shear, axial, and bending forces ( $f = k \cdot d$ ). Support reactions are calculated similarly.

Types of Structural Analysis

While a simple linear elastic analysis (PERFORM ANALYSIS) is the most common, STAAD offers several advanced, highly specialized analysis methods for complex structures:

Linear vs Non-Linear Analysis

Linear Static Analysis: Assumes structural deformations are very small and materials behave perfectly elastically (stress proportional to strain). The relationship between applied force and resulting displacement ( $[K]$ ) is assumed to be constant throughout the entire loading process. If you double the load, the deflection exactly doubles.
P-Delta Analysis (PDELTA ANALYSIS): A complex non-linear analysis method that explicitly accounts for secondary moments ( $P \times \Delta$ ). These secondary moments are caused when a large vertical axial load ( $P$ ) acts on a column that has already displaced laterally ( $\Delta$ ) due to wind or seismic forces. This "P-Delta effect" geometrically reduces the apparent lateral stiffness of the frame. This is absolutely critical for accurately analyzing tall, flexible, or highly slender structures.

The Mathematics of P-Delta Analysis

In a linear analysis, the global stiffness matrix

[K]

depends only on the material (

E

) and the original geometry (

A, I, L

). It is called the Elastic Stiffness Matrix

[K_E]

. In a P-Delta analysis, STAAD modifies this matrix iteratively.

It introduces a Geometric Stiffness Matrix

[K_G]

, which depends exclusively on the internal axial forces (

P

) currently within the members, not their material properties. A compressive axial load creates a "negative stiffness" (it wants to buckle), reducing the column's lateral resistance.

P-Delta Modified Stiffness Matrix

The effective stiffness matrix used in an iterative P-Delta non-linear analysis.

\left( [K_E] - [K_G] \right) \{D\} = \{F\}

Variables

Symbol	Description	Unit
$[K]_{eff}$	Effective total stiffness matrix	-
$[K_E]$	Standard elastic stiffness matrix (based on E, I, A)	-
$[K_G]$	Geometric stiffness matrix (based purely on internal axial force P and length L)	-
$\{D\}$	Unknown global displacements	-
$\{F\}$	Applied external loads	-

Because

[K_G]

depends on the internal force

P

, which in turn depends on the displacement

\{D\}

, the equation cannot be solved in one step. STAAD must iterate: calculate forces, update

[K_G]

, solve for new displacements, update forces again, until the displacements converge to a stable value.

Crucial Pre-Analysis Checks

Before blindly clicking "Run Analysis," an experienced engineer must perform rigorous error checking:

Multiple Structures Check: Carefully ensure the model isn't accidentally split into two or more separate, physically unconnected parts. If present, STAAD will warn you in the .ANL file.
Missing Properties/Materials: Verify if every single geometric member in the model has a defined cross-section and assigned material property ( $E$ ). The matrix $[K]$ will contain zeros and the analysis will fatally fail without these.
Check for Instabilities: Crucially check for zero stiffness conditions (singular matrices). For example, a simple 2D frame modeled entirely with pinned connections (member releases) and only roller supports will be fundamentally unstable and "fly away" under any applied lateral load, causing the Cholesky solver to fail.

Post-Processing: Reviewing Results

Visualizing how the structure behaves after mathematical computation.

After a successful analysis run, the engineer switches the interface to the Post-Processing mode. This is where the mathematical data is visualized. Use the interactive simulation below to visualize the relationship between applied loads on a simple beam and the resulting crucial design diagrams.

Post-Processing Results Visualizer

Load Magnitude (P or w)10 kN

Max Bending Moment25.0 kN·m

Max Shear Force5.0 kN

Beam Length10 m

Physical Model

Bending Moment Diagram (BMD)

Shear Force Diagram (SFD)

Key Visualizations

Deflection and Displacements:

The very first thing an engineer should check is the exaggerated displaced shape of the structure. Does it make physical sense?

Global vs. Local Deflection

Node Displacements (Global): The precise translational ( $X, Y, Z$ ) and rotational ( $RX, RY, RZ$ ) movements of specific key joints relative to the origin. This represents the absolute sway or drift of the building, checking limits like $H/400$ .
Member Deflection (Local): The elastic deflection curve along a specific beam's length relative to an imaginary straight line connecting its two ends. This is crucial for verifying serviceability limits (e.g., span/ $360$ ) for individual floor beams.

Member Internal Forces (BMD and SFD):

This is the core, essential output required for actual member design.

Bending Moment Diagrams (BMD): Critical for designing main flexural reinforcement in concrete beams or selecting W-section sizes. Often denoted as $M_z$ (moment causing bending about the member's local Z-axis).
Shear Force Diagrams (SFD): Crucial for designing transverse stirrups in concrete or explicitly checking web shear capacity. Typically denoted as $F_y$ (shear force acting parallel to the member's local Y-axis).
Axial Force Diagram ( $F_x$ ): Important for designing columns and analyzing truss members in pure tension or compression.

Plate Stresses and Cut Lines:

For complex models utilizing finite plate elements, engineers must interpret complex color-coded stress contours.

Principal Stresses ( $S_{MAX}$ , $S_{MIN}$ ): The maximum and minimum normal stresses acting on a specific principal plane within the element.
Von Mises Stress: A complex combined yield criterion used to check the overall safety margin of highly ductile materials like steel plates under multi-axial stress states.
Cut Lines: A highly useful post-processing tool. An engineer can draw an arbitrary straight line across a colored stress contour map, and STAAD will extract the specific internal forces ( $M_X, M_Y$ ) along that exact line, converting contour colors back into a readable 2D graph for manual rebar calculations.

Interpreting the Output File (.ANL)

The primary STAAD output file (typically a .ANL extension) is a highly detailed, plain text document summarizing the entire analysis process. It importantly contains:

Crucial Output Contents

The Statics Check: This is the most critical part of the output file. It rigorously compares the total applied external loads versus the total calculated support reactions. These values must mathematically balance exactly (summing to zero). If they do not match, the entire analysis is fundamentally invalid.
Warnings and Errors: Any fatal errors (which stop the analysis, usually matrix singularities) or critical warnings (e.g., "Warning: Instability detected at node 532"). You must investigate and resolve every instability warning.
Print Output: The detailed numerical results of any explicitly requested PRINT commands included in the STAAD editor (e.g., PRINT MEMBER FORCES ALL).

Key Takeaways

The standard PERFORM ANALYSIS command conducts a linear static analysis by assembling and solving the Global Stiffness Matrix equation $[K]\{D\} = \{F\}$ .
A more complex PDELTA ANALYSIS is strictly required for slender structures. It iteratively modifies the stiffness matrix $([K_E] - [K_G])$ to account for secondary moments caused by axial loads acting on displaced nodes.
Prior to analysis, always ensure structural connectivity (no multiple structures), complete property assignments (preventing zero stiffness), and adequate support conditions.
An engineer must always thoroughly check the .ANL text output file for fatal errors, investigate stability warnings, and crucially verify the statics check (total applied load must exactly equal total reactions).
The Post-Processing interface visually distinguishes between Global Node Displacements (sway/drift) and Local Member Deflection (sag).
Unlike line elements, plate elements output complex stress contours. Cut Lines allow engineers to extract specific numerical values across these 2D surfaces for design.

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