--- title: "Derive beam element stiffness matrix" subject: "Finite Element Method" chapter: 3 key: "semester_8/finite_element_method:beam-element-stiffness-derivation" source: https://www.bctengineer.com/q/semester_8/finite_element_method/beam-element-stiffness-derivation site: BCT Engineering --- # Derive beam element stiffness matrix Finite Element Method · Chapter 3: Direct Stiffness Method: Discrete Finite Elements - Asked in: year not given, 8 marks - Video: [Coefficients of the stiffness matrix - Derivation - Beam element - CORRECTED & EXPANDED](https://www.youtube.com/watch?v=RNIZ8Ju3_CI) — TM'sChannel - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_8/finite_element_method/beam-element-stiffness-derivation - Whole chapter: https://www.bctengineer.com/md/semester_8/finite_element_method/chapter-3 ## Question Derive the stiffness matrix of a two-node beam element of length $L$ and flexural rigidity $EI$ having a transverse displacement and a rotation at each node. State the meaning of any one column of the matrix. ## Short answer Using Hermite cubic interpolation, the strain energy integral yields the symmetric stiffness matrix k = (EI/L³)[12 6L -12 6L; 6L 4L² -6L 2L²; -12 -6L 12 -6L; 6L 2L² -6L 4L²]. Column 1 represents the forces/moments for unit transverse displacement v₁=1 with all other DOFs zero: shear 12EI/L³ and moment 6EI/L² at node 1, and shear -12EI/L³ and moment 6EI/L² at node 2. ## Answer A beam element has 4 degrees of freedom: $\{d\} = \{v_1,\ \theta_1,\ v_2,\ \theta_2\}^T$ with corresponding nodal forces $\{f_1,\ m_1,\ f_2,\ m_2\}^T$. ```text v1,f1 v2,f2 | (th1,m1) | (th2,m2) o===================o 1 L 2 ``` #### Displacement function Since there are 4 nodal values, a cubic is chosen: $v(x) = a_1 + a_2x + a_3x^2 + a_4x^3$. Applying $v(0) = v_1$, $v'(0) = \theta_1$, $v(L) = v_2$, $v'(L) = \theta_2$ and solving gives the shape functions (Hermite cubics): $$ \begin{aligned} N_1 &= 1 - \frac{3x^2}{L^2} + \frac{2x^3}{L^3}, & N_2 &= x - \frac{2x^2}{L} + \frac{x^3}{L^2} \\ N_3 &= \frac{3x^2}{L^2} - \frac{2x^3}{L^3}, & N_4 &= -\frac{x^2}{L} + \frac{x^3}{L^2} \end{aligned} $$ #### Strain energy and stiffness Curvature is $\kappa = \dfrac{d^2v}{dx^2} = [B]\{d\}$, where $$ [B] = \left[\frac{-6}{L^2} + \frac{12x}{L^3},\ \ \frac{-4}{L} + \frac{6x}{L^2},\ \ \frac{6}{L^2} - \frac{12x}{L^3},\ \ \frac{-2}{L} + \frac{6x}{L^2}\right] $$ The strain energy is $U = \dfrac{1}{2}\displaystyle\int_0^L EI\,\kappa^2\,dx$, which gives $$ [k] = EI\int_0^L [B]^T[B]\,dx $$ Evaluating the integrals term by term (for example $k_{11} = EI\int_0^L \left(\frac{12x}{L^3} - \frac{6}{L^2}\right)^2dx = \frac{12EI}{L^3}$): $$ [k] = \frac{EI}{L^3}\begin{bmatrix} 12 & 6L & -12 & 6L \\ 6L & 4L^2 & -6L & 2L^2 \\ -12 & -6L & 12 & -6L \\ 6L & 2L^2 & -6L & 4L^2 \end{bmatrix} $$ #### Meaning of a column Column 1 is the set of forces and moments needed to give a unit deflection $v_1 = 1$ at node 1 while $\theta_1 = v_2 = \theta_2 = 0$: shear $12EI/L^3$ and moment $6EI/L^2$ at node 1, and the opposite shear $-12EI/L^3$ and moment $6EI/L^2$ at node 2. The matrix is symmetric and singular (rigid-body motion is possible until supports are applied). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why is a cubic polynomial chosen for the beam element? A. The element has 4 nodal degrees of freedom: v1, theta1, v2, theta2 B. The beam is 3 m long C. The curvature is constant D. Two nodes need three terms Answer: A. Four nodal values need four polynomial coefficients, so a cubic is used. (Key point: Degrees of freedom) 2. Which expression gives the beam element stiffness matrix? A. [k] = AE integral of [B]^T[B] dx with first derivatives B. [k] = EI integral of [B]^T[B] dx, where [B] is the second derivative of the shape functions C. [k] = integral of [N]^T[N] dx D. [k] = EI [B]/L Answer: B. Curvature is d2v/dx2 = [B]{d}, and the strain energy 1/2 integral of EI kappa^2 gives [k] = EI integral [B]^T [B] dx. (Key point: Strain energy) 3. What is the k11 term of the beam element stiffness matrix? A. 4EI/L B. 6EI/L^2 C. 12EI/L^3 D. 2EI/L Answer: C. k11 = EI integral of (12x/L^3 - 6/L^2)^2 dx = 12EI/L^3. (Key point: Matrix terms) 4. What does column 1 of the matrix represent? A. The displacements caused by a unit load B. The strain energy C. The shape functions D. Forces and moments needed for unit deflection v1 = 1 with the other three DOF zero Answer: D. Each column holds the nodal forces and moments required to give a unit value of one DOF while the others are zero. 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