4.4 Determinate Structures-1

4.4 Determinate Structures-1

Introduction to Structural Analysis

• The analysis of any structure begins with determining its stability and determinacy. This classification dictates the methods required to find internal forces and deflections.

• Determinate structures are those where all support reactions and internal member forces can be found using only the equations of static equilibrium.

• This section covers the criteria for determinacy, introduces powerful energy-based methods for calculating deflections, and applies these concepts to beams and simple frames.

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1. Degree of Determinacy

1.1 Classifying Structures

• The degree of determinacy indicates whether a structure is stable, and if so, whether equilibrium equations alone are sufficient for its analysis.

1.2 Determinacy of Trusses (Plane)

• For a plane truss, the condition is based on the number of members ($\boldsymbol{m}$), joints ($\boldsymbol{j}$), and reaction components ($\boldsymbol{r}$).

• Criteria:

  • $\boldsymbol{m + r = 2j}$: Statically Determinate (Equilibrium sufficient)

  • $\boldsymbol{m + r > 2j}$: Statically Indeterminate (Indeterminate to the $\boldsymbol{(m+r-2j)}$ degree)

  • $\boldsymbol{m + r < 2j}$: Unstable (Mechanism)

1.3 Determinacy of Beams and Frames (Plane)

• For beams and rigid frames, the condition involves the number of possible internal force components at critical sections and the reaction components.

• Each member can potentially have three internal forces: Axial Force ($\boldsymbol{N}$), Shear Force ($\boldsymbol{V}$), and Bending Moment ($\boldsymbol{M}$).

• General Formula:

  • Let $\boldsymbol{D_s}$ be the degree of static indeterminacy.

  • For a structure with $\boldsymbol{m}$ members, $\boldsymbol{j}$ joints (including supports treated as joints), and $\boldsymbol{r}$ reaction components:

    $\boldsymbol{D_s = (3m + r) - 3j}$

  • For beams, which are a single member with no internal releases, this simplifies. A beam has one member ($\boldsymbol{m=1}$). Let the number of support reaction components be $\boldsymbol{r}$. The formula becomes:

    $\boldsymbol{D_s = r - 3}$

• Interpretation of Result:

  • $\boldsymbol{D_s = 0}$: Statically Determinate

  • $\boldsymbol{D_s > 0}$: Statically Indeterminate (to the $\boldsymbol{D_s}$ degree)

  • $\boldsymbol{D_s < 0}$: Unstable

• Alternative Check for Beams: A beam is determinate if it has exactly three unknown reaction components, provided they are not parallel or concurrent.

1.4 Importance of Determinacy

• Determinate Structures: Reactions and internal forces are independent of member material ($\boldsymbol{E}$) and cross-section ($\boldsymbol{A, I}$). Analysis is straightforward.

• Indeterminate Structures: Internal forces depend on the relative stiffness ($\boldsymbol{EI, EA}$) of members. Requires compatibility conditions in addition to equilibrium.

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2. Energy Methods

• Energy principles provide elegant and powerful techniques, especially for finding deflections in complex structures. They are based on the law of conservation of energy.

2.1 Strain Energy (U)

• Definition: The energy stored within a deformable structure due to the work done by internal stresses as the structure deforms.

• Expressions for Common Actions:

  • Axial Force: For a member of length $\boldsymbol{L}$, constant axial load $\boldsymbol{N}$, area $\boldsymbol{A}$, and modulus $\boldsymbol{E}$.

    $\boldsymbol{U_N = \int_0^L \frac{N^2}{2EA} \, dx \quad \text{or} \quad U_N = \frac{N^2 L}{2EA}}$

  • Bending Moment: For a member of length $\boldsymbol{L}$, varying moment $\boldsymbol{M(x)}$, flexural rigidity $\boldsymbol{EI}$.

    $\boldsymbol{U_M = \int_0^L \frac{M^2}{2EI} \, dx}$

  • Shear Force: Often negligible for slender beams. For uniform shear $\boldsymbol{V}$, area $\boldsymbol{A}$, shape factor $\boldsymbol{k}$, shear modulus $\boldsymbol{G}$.

    $\boldsymbol{U_V = \int_0^L \frac{kV^2}{2GA} \, dx}$

2.2 Principle of Virtual Work

• Fundamental Concept: A cornerstone method for calculating displacements. It equates the external virtual work done by fictitious (virtual) forces moving through real displacements to the internal virtual work done by virtual internal forces moving through real deformations.

  $\boldsymbol{1 \cdot \Delta = \sum \int (\text{virtual internal stress}) \times (\text{real strain}) \, dV}$

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3. Virtual Work Method for Deflection

• This is the most versatile application of the Principle of Virtual Work for linear elastic structures.

3.1 Procedure

1. Real System: Analyze the structure under the given real loads to find the real internal force distributions ($\boldsymbol{N, M, V}$).

2. Virtual System: Remove all real loads. Apply a unit virtual load (a dimensionless force = 1 or moment = 1) at the point and in the direction of the desired displacement/rotation.

3. Analyze Virtual System: Find the internal forces ($\boldsymbol{n, m, v}$) caused by this unit virtual load.

4. Apply Virtual Work Equation: The desired deflection/rotation $\boldsymbol{\Delta}$ is given by:

   $\boldsymbol{1 \cdot \Delta = \sum \int_{0}^{L} \frac{n N}{EA} \, dx + \sum \int_{0}^{L} \frac{m M}{EI} \, dx + \sum \int_{0}^{L} \frac{k v V}{GA} \, dx}$

   For most beams and frames, axial and shear deformations are negligible compared to bending. The formula simplifies to:

   $\boldsymbol{\Delta = \sum \int_{0}^{L} \frac{m M}{EI} \, dx}$

3.2 Key Points

• The virtual load is fictitious and is applied only to help compute the displacement.

• The real system contains the actual loads and resulting deformations.

• Superposition can be used: The displacement due to multiple real loads is the sum of the integrals for each load case.

• Sign Convention: Consistent sign convention for $\boldsymbol{m}$ and $\boldsymbol{M}$ is crucial. Positive $\boldsymbol{\Delta}$ indicates displacement in the direction of the applied unit virtual load.

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4. Application to Deflection of Beams and Portal Frames

4.1 Deflection of Determinate Beams

• The Virtual Work Method reduces to evaluating a single integral for the beam segment.

  $\boldsymbol{\Delta = \int_{0}^{L} \frac{m(x) M(x)}{EI} \, dx}$

• Process:

  1. Determine $\boldsymbol{M(x)}$ for the real beam under actual loads.

  2. Determine $\boldsymbol{m(x)}$ for the same beam under a unit load at the point of interest.

  3. For prismatic beams (constant $\boldsymbol{EI}$), compute $\boldsymbol{\int m M \, dx}$.

  4. Divide by $\boldsymbol{EI}$ to get the deflection.

• Tabular Integration: For beams with piecewise linear/quadratic moment diagrams, the integral can be efficiently computed using the product integral table (e.g., area of real diagram times ordinate of virtual diagram at centroid).

4.2 Deflection of Determinate Portal Frames

• Portal frames are common planar structures with columns and a beam (rigidly or pin-connected).

• Analysis Steps:

  1. Real System Analysis: Perform static analysis of the frame under real loads to find the $\boldsymbol{M}$ diagram for each member (columns and beam).

  2. Virtual System Analysis: Apply a unit virtual load at the desired location/direction. Re-analyze the frame to find the corresponding $\boldsymbol{m}$ diagram.

  3. Integrate per Member: The total displacement is the sum of integrals over all members.

     $\boldsymbol{\Delta = \sum_{\text{members}} \int_{\text{member}} \frac{m M}{EI} \, dx}$

  4. Account for member properties: Use the appropriate $\boldsymbol{EI}$ for each member if they differ.

• Considerations for Frames:

  • Axial deformation in columns may sometimes be significant if the frame is slender or the deflection sought is horizontal at the top.

  • Typically, only bending contributions are considered for lateral deflections and joint rotations in standard rigid frames.

4.3 Finding Rotations

• To find the rotation (slope) at a point, apply a unit virtual couple (moment = 1) at that point in the virtual system. The procedure is identical, and the result $\boldsymbol{\theta}$ is given by the same integral formula.

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Summary

• The degree of static indeterminacy ($\boldsymbol{D_s}$) must be checked first. For determinate structures ($\boldsymbol{D_s=0}$), reactions and internal forces are found using statics alone.

• Energy methods, particularly the Principle of Virtual Work, provide a general and powerful technique for calculating deflections and rotations.

• The Virtual Work Method involves:

  1. Analyzing the structure under real loads (for $\boldsymbol{M}$).

  2. Analyzing it under a unit virtual load/couple at the desired location (for $\boldsymbol{m}$).

  3. Evaluating the integral $\boldsymbol{\Delta = \int (m M / EI) \, dx}$ over all members.

• This method is systematically applied to find deflections in determinate beams and portal frames, with bending deformation usually being the dominant contributor.