A Minimum Energy Principle with Variable Loads and Its Application in Structural Optimization

China Railway Eryuan Engineering Group Co., Ltd.
247073858@qq.com

June 17, 2025

Abstract

In mechanical structures, conventional energy principles are formulated for fixed loads. However, in engineering structures such as bridges, loads often need to be adjusted to achieve reasonable internal force distribution and material savings. Nevertheless, theoretical research in this area remains limited. To address this problem, this paper establishes a constrained functional extremum problem and proposes a minimum energy principle with variable loads in mechanics: the system energy is minimized when the virtual work of variable loads equals zero, i.e., when the variable loads are orthogonal to their corresponding displacements. It is demonstrated that the principle of minimum potential energy is a special case of this principle. Furthermore, the orthogonality condition can serve as a supplementary equation to the fundamental equations of mechanics. This paper also presents specific computational methods for applying this theory to structural optimization: the forced displacement method, the large stiffness method, and the generalized unit load method. This theory transforms the control of variable loads into displacement control, enabling convenient solution using tools such as the finite element method. Several structural examples are presented to demonstrate the broad practical value of the proposed theory.

Keywords: minimum energy principle, orthogonality, variable loads, bridges, structural optimization

In engineering structures, active load control is often required to ensure reasonable structural behavior and economic efficiency. This need is particularly urgent in structural engineering, such as: determining prestressing tendon layouts for reasonable internal force distribution in prestressed concrete beams; optimizing cable forces in cable-stayed bridges for optimal structural performance; adjusting hanger forces in arch bridges and tie forces in tied-arch bridges; and controlling tension forces in spatial cable-net structures. The influence matrix method is generally employed and remains dominant, while intelligent algorithms and data-driven techniques have grown rapidly in recent years. Fleming (1979) \cite{Fleming1979} proposed the influence matrix method, establishing its foundation. Wang P.H., et al. (1993) \cite{Wang1993} developed a linear programming model for cable force optimization in cable-stayed bridges based on the influence matrix. Xiao Rucheng (1997) \cite{Xiao1997} introduced a bending energy minimization criterion based on the influence matrix to optimize cable forces and minimize the bending potential energy of cable-stayed bridge girders. Tibert (2002) \cite{Tibert2002} proposed the force density method for form-finding of spatial cable-net structures. Li Yongle (2010) \cite{Li2010} presented a genetic algorithm-based optimization of hanger forces in tied-arch bridges with a multi-objective fitness function. Zhang et al. (2021) \cite{Zhang2021} proposed a hybrid PSO-GA algorithm that combines the global search capability of PSO with the local optimization ability of GA, significantly improving computational efficiency. Wang Hao (2023) \cite{Wang2023} developed an intelligent control method for prestressed steel structures driven by digital twins. Li Hui (2022) \cite{Li2022} developed an LSTM prediction model that can predict cable force variations in cable-stayed bridges. All these optimization problems can be归结为variable load optimal control problems.

Energy principles are the cornerstone of structural analysis. The principle of virtual displacements was proposed by John Bernoulli in 1717, and Maxwell's reciprocal theorem was established in 1864. In 1872, Betti generalized the displacement reciprocity theorem to the work reciprocity theorem. Castigliano proposed his first and second theorems in 1879, and Engesser introduced the complementary energy method in 1889. In 1950, Reissner proposed a generalized variational principle for two types of variables in elasticity \cite{Reissner1950}, and in 1954, Hu Haichang proposed a generalized variational principle for three types of variables \cite{Hu1954}. In 1983, Long Yuqiu proposed a partitioned mixed generalized variational principle \cite{Long1983}. All these structural energy principles were derived for fixed loads. The applicability of these principles to structural optimization problems with variable loads requires further investigation.

Energy principles in structural analysis can all be derived using variational methods, which essentially involve solving functional extremum problems. Orthogonality conditions hold a central position in functional extremum problems, contact mechanics, and variational methods. Courant & Hilbert (1953) first rigorously proved the orthogonality of constrained variational problems, proposing dual-space orthogonality and laying the mathematical foundation for energy orthogonality principles in elasticity (such as modal analysis of beams) \cite{Courant1953}. Lions & Stampacchia (1967) established a general framework for variational inequalities, proving that the solution $u$ and multiplier $\lambda$ satisfy complementary orthogonality conditions, and noted that orthogonality is essentially an expression of Hilbert's projection theorem \cite{Lions1967}. Duvaut & Lions (1976) formulated the Signorini contact problem as a variational inequality, rigorously proving the orthogonality between contact pressure $\lambda$ and displacement gap $u - g$, and introduced the concept of dual pairing to explain the physical meaning of orthogonality as "contact forces do work only in the effective contact zone" \cite{Duvaut1976}. Oden & Reddy (1983) \cite{Oden1983} systematically discussed orthogonality modifications under non-homogeneous boundary conditions, noting that non-homogeneous boundaries require additional correction terms to the orthogonality condition. Kikuchi & Oden (1988) proposed orthogonality conditions for contact problems \cite{Kikuchi1988}. Brezzi & Fortin (1991) proposed the inf-sup condition to guarantee numerical stability of orthogonality in mixed finite element methods, solving non-standard orthogonality problems (such as Stokes equations) and providing theoretical tools for non-matching mesh discretization in contact problems \cite{Brezzi1991}. Reddy (2002) \cite{Reddy2002} systematically discussed orthogonality under non-homogeneous boundaries, specifically proving for beam differential equations that weighted orthogonality between constraint forces $q(x)$ and displacements $w(x)$ is a necessary condition for energy extremum. Wriggers (2002) extended orthogonality to frictional contact: tangential friction forces $\lambda_t$ and slip $s$ satisfy $\lambda_t \cdot s \, ds = 0$ \cite{Wriggers2002}. Arnold et al. (2005) used discontinuous Galerkin methods to preserve discrete orthogonality on non-conforming meshes, resolving numerical oscillations in Mortar methods and proving that orthogonality error relates to mesh size $h$ and polynomial degree $p$ as $O(h^{p+1/2})$ \cite{Arnold2005}.

This paper employs variational methods and utilizes orthogonality conditions for functional extremum to extend the minimum energy principle in mechanics to cases with variable loads and applies it to structural optimization design.

2.1 Problem Description

Consider the following functional extremum problem:
$$
J[w, f] = \int \left[EI(w'')^2 + \lambda_1(x)(q(x) - C_1)\right]dx + \int \left[EI(w'')^2 + EA(u')^2 + q_0w + qw + f_0u + fu\right]dx + \int \lambda_2(x)(f(x) - C_2)dx
$$
where:
- $w(x)$ is the vertical displacement field (free variable)
- $u(x)$ is the axial displacement field (free variable)
- $q(x)$ is the constraint force satisfying $q(x) < C_1$ (for all $x \in [0, 1]$)
- $f(x)$ is the constraint force satisfying $f(x) < C_2$ (for all $x \in [0, 1]$)
- $EI$ is flexural stiffness, $EA$ is axial stiffness
- $q_0$ is distributed vertical load (constant), $f_0$ is distributed axial load (constant)

KKT conditions include:
- Primal feasibility: $q(x) \leq C_1, f(x) \leq C_2$
- Dual feasibility: $\lambda_1(x) \geq 0, \lambda_2(x) \geq 0$
- Complementary slackness: $\lambda_1(x)(q(x) - C_1) = 0, \lambda_2(x)(f(x) - C_2) = 0$
- Extremum condition: Variation of functional $L$ with respect to $w, u, q, f$ equals zero.

2.2 Variational Derivation

Taking variation with respect to $w$:
$$
\delta J = \int \left[EIw''\delta w'' + q_0\delta w + q\delta w\right]dx
$$
Applying integration by parts twice:
$$
\int EIw''\delta w''dx = \left[EIw''\delta w'\right]_0^L - \left[EIw'''\delta w\right]_0^L + \int EIw''''\delta w dx
$$
Thus the extremum condition yields the Euler-Lagrange equation:
$$
EIw'''' + q_0 + q = 0
$$
with boundary conditions:
$$
w''(0) = w''(1) = 0, \quad w'''(0) = w'''(1) = 0
$$

Similarly for $u(x)$:
$$
-EAu'' + f_0 + f = 0
$$

2.3 Proof of Orthogonality Between Variable Loads and Displacements

For a general function $q(x)$, expand using basis functions:
$$
q(x) = \sum a_k\psi_k(x)
$$
where ${\psi_k(x)}$ are basis functions (e.g., polynomials or trigonometric functions).

Optimization condition:
$$
\int (w + \lambda_1)\psi_k dx = 0 \quad \forall k
$$
When $q = \sum Q_i\delta(x - L_i)$ represents concentrated loads, based on the properties of Dirac functions:
$$
\int q(w + \lambda_1)dx = \sum Q_i(w_i + \lambda_{1i}) = 0
$$
where $Q_i$ satisfies constraint function properties. When unconstrained, $w_i + \lambda_{1i} = 0$.

For a general function $f(x)$, expand using basis functions:
$$
f(x) = \sum a_k\phi_k(x)
$$
where ${\phi_k(x)}$ are basis functions.

Optimization condition:
$$
\int (u + \lambda_2)\phi_k dx = 0 \quad \forall k
$$
When $f = \sum F_i\delta(x - L_i)$ represents concentrated loads:
$$
\int f(u + \lambda_2)dx = \sum F_i(u_i + \lambda_{2i}) = 0
$$
where $Q_i$ satisfies constraint function properties. When unconstrained, $u_i + \lambda_{2i} = 0$.

In the special case when $q, f$ are not subject to inequality constraints, the orthogonality conditions simplify to:
$$
\int qw\,dx = 0, \quad \int fu\,dx = 0
$$
If $q, f$ are constants, they can be taken outside the integral:
$$
\int w\,dx = 0, \quad \int u\,dx = 0
$$

Based on the above orthogonality conditions, when a structure is subjected to variable loads, minimizing structural energy requires that the virtual work (also called external work) done by variable loads equals zero. Specifically, when variable loads are constant, the integral of displacements at the load locations must be zero. This orthogonality condition transforms the problem of solving for optimal variable loads into a displacement problem, facilitating computational implementation, especially when combined with finite element displacement conditions.

1. Problem Description: Simply Supported Beam Under Distributed Load

Consider bending of a simply supported beam under distributed load with governing equation:
$$
EIw''''(x) + q(x) = 0
$$
where:
- $w(x)$ is the beam deflection function
- $EI$ is flexural rigidity (constant)
- $q(x) = q_0 + ax$ is linearly distributed load

2. Differential Equation Formulation

Rewrite the governing equation as:
$$
w''''(x) = -\frac{q(x)}{EI} = -\frac{q_0 + ax}{EI}
$$

3. Successive Integration

First integration (third derivative):
$$
w'''(x) = -\frac{1}{EI}\int(q_0 + ax)dx + C_1 = -\frac{1}{EI}\left(q_0x + \frac{ax^2}{2}\right) + C_1
$$

Second integration (second derivative, i.e., bending moment):
$$
w''(x) = -\frac{1}{EI}\left(\frac{q_0x^2}{2} + \frac{ax^3}{6}\right) + C_1x + C_2
$$

Third integration (first derivative, i.e., rotation):
$$
w'(x) = -\frac{1}{EI}\left(\frac{q_0x^3}{6} + \frac{ax^4}{24}\right) + \frac{C_1x^2}{2} + C_2x + C_3
$$

Fourth integration (deflection):
$$
w(x) = -\frac{1}{EI}\left(\frac{q_0x^4}{24} + \frac{ax^5}{120}\right) + \frac{C_1x^3}{6} + \frac{C_2x^2}{2} + C_3x + C_4
$$

4. Boundary Conditions

For a simply supported beam:
- Zero deflection: $w(0) = 0$ and $w(L) = 0$
- Zero bending moment: $w''(0) = 0$ and $w''(L) = 0$

Applying boundary conditions:
$w(0) = C_4 = 0 \Rightarrow C_4 = 0$
$w''(0) = C_2 = 0 \Rightarrow C_2 = 0$
$w''(L) = -\frac{1}{EI}\left(\frac{q_0L^2}{2} + \frac{aL^3}{6}\right) + C_1L = 0$
$w(L) = -\frac{1}{EI}\left(\frac{q_0L^4}{24} + \frac{aL^5}{120}\right) + \frac{C_1L^3}{6} + C_3L = 0$

5. Deflection Function Expression

Substituting constants into deflection expression:
$$
w(x) = -\frac{1}{EI}\left(\frac{q_0x^4}{24} + \frac{ax^5}{120}\right) + C_3x
$$
with $C_3 = -\frac{q_0L^3}{24EI} - \frac{aL^4}{120EI}$

6. Additional Constraint Condition

According to problem requirements, deflection must satisfy:
$$
\int_0^L xw(x)dx = 0
$$
Substituting $w(x)$ expression and integrating term by term yields the condition for parameter $a$.

3 Finite Element Discretization and Proof

Previous chapters presented theoretical analysis of the minimum virtual work principle. For engineering applications, finite element discretization can be employed, as detailed below. The finite element discretization process is conventional and will not be analyzed in detail here. For convenience, matrix notation is adopted. After discretization, the finite element equation becomes:
$$
Ku = F + Eq
$$
where $u, F$ are $n$-element column vectors, $K$ is an $n \times n$ matrix, $q$ is a control load vector with $m$ elements ($m \leq n$), and $E$ is a position expansion matrix with elements of 0 and 1 to incorporate control loads $q$ into the finite element equilibrium equations.

Assume control loads $q$ have $m_1$ equality constraints:
$$
h(q) = 0
$$
where $h$ is an $m_1$-element vector function and $0$ represents $[0, 0, ..., 0]^T$. Additionally, there are $p$ inequality constraints:
$$
g(q) \leq 0
$$
where $C = [c_1, c_2, ..., c_p]^T$ is a constant vector.

Since $q$ is an $m$-dimensional vector with $m_1$ constraints, its dimension can be reduced to $m_2 = m - m_1$, meaning there are only $m_2$ independent variables:
$$
q = [q_1, q_2, ..., q_{m_1}, q_{m_1+1}, ..., q_m]^T
$$
with $m_2$ independent variables $[q_{m_1+1}, q_{m_1+2}, ..., q_m]^T$ and dependent variables $[q_1, q_2, ..., q_{m_1}]^T$.

The $m_1$ equality constraints can be expressed in terms of independent variables:
$$
q_1 = f_1(q_{m_1+1}, q_{m_1+2}, ..., q_m) \
q_2 = f_2(q_{m_1+1}, q_{m_1+2}, ..., q_m) \
\vdots \
q_{m_1} = f_{m_1}(q_{m_1+1}, q_{m_1+2}, ..., q_m)
$$

Based on structural virtual work and introducing Lagrange multipliers, the virtual work expression becomes:
$$
J = (F + Eq)^Tw - \frac{1}{2}w^TKw + \mu^Th + \lambda^Tg
$$
where $E$ is the load position expansion matrix. Taking variation:
$$
\delta J = (\delta w)^T[F + Eq - Kw] + (\delta q)^T[E^Tw + \text{(cid:18)(cid:19)}^T\lambda] = 0
$$

Analyzing the $\delta w$ term: since $\delta w$ is arbitrary, the equilibrium equation is obtained:
$$
Kw = F + Eq
$$

Analyzing $\delta q$: since $\delta q$ is an $m$-dimensional vector with $m_1$ constraints, it can be reduced to $m_2 = m - m_1$ independent variables. Therefore, we cannot directly set $\delta q = 0$ to obtain the control equation. Let:
$$
q = [q_1, q_2, ..., q_{m_1}, q_{m_1+1}, ..., q_m]^T
$$
Denote independent variables as $[q_{m_1+1}, q_{m_1+2}, ..., q_m]^T$ and dependent variables as $[q_1, q_2, ..., q_{m_1}]^T$.

The $m_1$ equality constraints can be expressed as:
$$
f_1 = f_1(q_{m_1+1}, q_{m_1+2}, ..., q_m) \
f_2 = f_2(q_{m_1+1}, q_{m_1+2}, ..., q_m) \
\vdots \
f_{m_1} = f_{m_1}(q_{m_1+1}, q_{m_1+2}, ..., q_m)
$$

Taking variation yields the Jacobian matrix of $q$ with respect to independent variables. Substituting into $\delta J$ gives:
$$
(\delta q)^T[E^Tw + \text{(cid:18)(cid:19)}^T\lambda] = (\delta\text{(cid:18)(cid:19)})^T[E^Tw + \text{(cid:18)(cid:19)}^T\lambda] = 0
$$
Since $\delta\text{(cid:18)(cid:19)}$ is arbitrary:
$$
\text{(cid:18)(cid:19)}^T[E^Tw + \text{(cid:18)(cid:19)}^T\lambda] = 0
$$
This is the control equation for minimizing system energy.

Additionally, according to KKT conditions, Lagrange multipliers $\lambda_i$ for inequality constraints must satisfy dual feasibility:
$$
\lambda \geq 0
$$
and complementary slackness:
$$
\lambda^Tg(q) = 0
$$

3.2 Orthogonality Proof for Control Loads and Displacements

Since $q$ is an $m$-dimensional column vector with $m_1$ constraints, its dimension reduces to $m_2 = m - m_1$. Thus $q$ has $m_2$ independent basis vectors $\phi_i, i = 1, 2..m_2$:
$$
q = \sum_{i=1}^{m_2} a_i\phi_i
$$

If constraints $g(q), h(q)$ are linear functions, let:
$$
g(q) = Aq - b = \sum_{i=m_2+1} q_i\phi_i - b \
\lambda^Tg(q) = \lambda^T(\sum q_i\phi_i - b) \
h(q) = Bq - c = \sum q_i\phi_i - c \
\mu^Th(q) = \mu^T(\sum q_i\phi_i - c)
$$

The energy $J$ can be viewed as a function of $q_i$. Substituting into the previous equation:
$$
J(q_i) = (F + E\sum q_i\phi_i)^Tw - \frac{1}{2}w^TKw + \mu^T(\sum q_i\phi_i - c) + \lambda^T(\sum q_i\phi_i - b)
$$

Taking derivative yields optimality condition:
$$
\frac{\partial J}{\partial q_i} = (E\phi_i)^Tw + \mu^T\phi_i + \lambda^T\phi_i = 0
$$

Expanding:
$$
(Eq)^Tw + \mu^Th(q) + \lambda^Tg(q) = \sum q_i(E\phi_i)^Tw + \mu^T(\sum q_i\phi_i - c) + \lambda^T(\sum q_i\phi_i - b) \
= \sum q_i[(E\phi_i)^Tw + \mu^T\phi_i + \lambda^T\phi_i] - \mu^Tc - \lambda^Tb = -\mu^Tc - \lambda^Tb
$$

If $b = 0, c = 0$:
$$
(Eq)^Tw + \mu^Th(q) + \lambda^Tg(q) = 0
$$
This is the orthogonality condition. If $b \neq 0, c \neq 0$, it is equivalent to applying constants $b, c$ to the structure first, then applying the orthogonality condition.

3.3 Special Cases and Case Analysis

Several special cases are presented to illustrate the derived equations:

1) When control loads $q$ are unconstrained, equation (51) simplifies to:
$$
E^Tw = 0
$$

2) When control loads $q$ have only equality constraints, equation (51) simplifies to:
$$
\text{(cid:18)(cid:19)}^T[E^Tw + \text{(cid:19)}^T\mu] = 0
$$

Example: For control loads $q = [q_1, q_2]$ at two nodes with constraint $h(q_1, q_2) = q_1 - q_2 = 0$, taking $q_2$ as independent variable gives:
$$
\text{(cid:18)(cid:19)} = [1, -1]
$$
Substituting into (58):
$$
\text{(cid:18)(cid:19)}^T = [1, 1] \
\text{(cid:18)(cid:19)} = w_1 + w_2 = 0
$$

This indicates that when loads at two nodes are equal, minimizing system energy requires equal displacements at these nodes, making the virtual work of control loads zero. This simple example can be applied to cable force adjustment in cable-stayed bridges and hanger force adjustment in arch bridges.

If the equality constraint is $h(q_1, q_2) = q_1 + q_2 = 0$, the orthogonality condition gives $w_1 - w_2 = 0$, again making virtual work zero: $q_1w_1 + q_2w_2 = 0$.

If the equality constraint is $h(q_1, q_2) = q_1 - 2q_2 = 0$, the orthogonality condition becomes $2w_1 + w_2 = 0$, with zero virtual work: $q_1w_1 + q_2w_2 = 0$.

These three simple cases demonstrate that system energy is minimized when virtual work of control loads equals zero, validating the proposed minimum virtual work principle.

3) When control loads $q$ have only inequality constraints, equation (51) simplifies to:
$$
E^Tw + \text{(cid:19)}^T\lambda = 0
$$

Example: With single inequality constraint $q < q_0$, i.e., $g(q) = q - q_0 \leq 0$.

4 Algorithm and Application

4.1 Cantilever Beam with Constrained Concentrated Load at End

To verify the derived minimum energy principle with inequality constraints, consider a cantilever beam of length $L$ with fixed left end and free right end, subjected to uniform load $q$ (positive downward) and concentrated force $F$ at the free end (positive upward, with $F \leq C$). Find the variable load $F$ that minimizes structural elastic potential energy:
$$
U(F) = \int \frac{EI(w'')^2}{2}dx - \int qw(x)dx + Fw(L)
$$

According to the minimum potential energy principle, introduce Lagrange multiplier $\lambda$ to construct the generalized functional:
$$
\Pi[w] = \int \frac{EI(w'')^2}{2}dx - \int qwdx + Fw(L) + \lambda(F - C)
$$

Variation yields the Euler-Lagrange equation and orthogonality condition:
$$
EIw'''' = q
$$

Boundary conditions:
$$
w(0) = 0, \quad w'(0) = 0, \quad w''(L) = 0, \quad EIw'''(L) = F
$$

Orthogonality condition:
$$
w(L) + \lambda = 0
$$

From KKT conditions, complementary slackness:
$$
\lambda(F - C) = 0, \quad \lambda \geq 0, \quad F \leq C
$$

Solving differential equation $EIw'''' = q$:
$$
w(x) = \frac{qx^4}{24EI} + C_3x + C_4
$$

Applying boundary conditions yields deflection curve:
$$
w(x) = \frac{qx^4}{24EI} - \frac{qLx^3}{6EI} + \frac{(qL^2 - 2F) x^2}{4EI}
$$
$$
w(L) = \frac{3qL^4 - 8FL^3}{24EI}
$$

Potential energy expression:
$$
U(F) = \frac{1}{120EI}(3q^2L^5 - 8qFL^4 + 4F^2L^3)
$$

From complementary slackness, discuss cases:
- Case 1: Constraint inactive ($F < C, \lambda = 0$), with $w(L) = 0$, when $C \geq \frac{3qL}{8}$
- Case 2: Constraint active ($F = C, \lambda \geq 0$), with $\lambda = -w(L) = -\frac{3qL^4 - 8CL^3}{24EI} \geq 0$, when $C < \frac{3qL}{8}$

The optimal solution is:
$$
F^* = \begin{cases}
\frac{3qL}{8}, & \text{if } C \geq \frac{3qL}{8} \
C, & \text{if } C < \frac{3qL}{8}
\end{cases}
$$

Special cases:
- Unconstrained ($C \to +\infty$): reduces to classical solution $F = \frac{3qL}{8}$
- Fully constrained ($C = 0$): $F = 0$, only uniform load acts

[FIGURE:1]: The controlled load on the cantilever beam is a distributed load.

4.2 Cantilever Beam with Distribution Force Under Specific Function Constraint

This section verifies orthogonality for distributed forces under specific function constraints. The control load is a distributed force $f$. Find $f$ to minimize strain energy as shown in Figure 1.

According to equation (??), the Euler equation for the cantilever beam element is:
$$
d^2y'' - q = 0, \quad 0 \leq x < L_1 \
d^2y'' - q - f = 0, \quad L_1 < x < L
$$

Boundary conditions from (??)-(??):
$$
y(0) = 0, \quad y'(0) = 0, \quad y''(L) = 0, \quad y'''(L) = 0 \
y_1(x_1) = y_2(x_1), \quad y_1'(x_1) = y_2'(x_1), \quad y_1''(x_1) = y_2''(x_1)
$$

From control conditions obtained from (??):
$$
\int_{L_1}^L y_2 dx = 0
$$

This equation shows that internal energy is minimized when the displacement integral at control load $F$ equals zero, i.e., when virtual work done by $F$ is zero. This validates the proposed minimum virtual work principle for cantilever beams under distributed loads.

The differential equation solutions are:
$$
y_1(x) = \frac{6L^2f + 6L^2q - 6fx_1^2}{24EI}x^2 + \frac{qx^4}{24EI} - \frac{4Lfx^3}{24EI} - \frac{4Lqx^3}{24EI} + \frac{4fxx_1^3}{24EI}
$$
$$
y_2(x) = \frac{fx^4 + fx_1^4 + qx^4 - 4fxx_1^3 + 6L^2fx^2 + 6L^2qx^2 - 4Lfx^3 - 4Lqx^3}{24EI}
$$

Total virtual work:
$$
W(F) = \frac{1}{120EI}(3L^5f^2 + 6L^5fq + 3L^5q^2 - 10L^2f^2x_1^3 - 10L^2fqx_1^3 + 5Lf^2x_1^4 + 5Lfqx_1^4 + 2f^2x_1^5 - fqx_1^5)
$$

From control equation (??):
$$
\frac{\partial W}{\partial f} = \frac{1}{120EI}(6L^5f + 6qL^5f - 20L^2f^2x_1^3 - 10qL^2fx_1^3 + 10Lf^2x_1^4 + 5qLfx_1^4 + 4f^2x_1^5 - qfx_1^5) = 0
$$

Solving yields:
$$
f = -\frac{6L^5 - 10L^2x_1^3 + 5Lx_1^4}{6L^5 - 20L^2x_1^3 + 10Lx_1^4 + 4x_1^5}q
$$

The minimum potential energy is:
$$
U_{\min} = \frac{36L^3 - 28L^2x_1 + 8Lx_1^2}{480EI(3L^3 + 6L^2x_1 + 9Lx_1^2 - x_1^3 + 2x_1^3)}q^2
$$

Using extremum condition verification, $W'(F)$ yields the same result as above. For visualization with $E=1, I=1, q=1, L=40, x_1=20$, displacement, controlled load work, fixed load virtual work, and strain energy curves versus $F$ are shown in the figure.

As $f$ decreases, negative work by the controlled load increases and system internal energy decreases. At $F = -1.18$, $\int y_2 dx = 0$, controlled load work reaches minimum, and strain energy is minimized. When $f$ continues decreasing, $y_L$ changes from positive to negative (downward to upward deformation), controlled load begins positive work, total work increases, and internal energy rises. At $f = -2.36$, controlled load effects exceed fixed load effects and adverse effects appear.

In summary, the proposed minimum virtual work principle is correct and can effectively calculate minimum potential energy (minimum sum of squared moments) to obtain optimal control loads and maximum material savings.

[FIGURE:2]: Displacement and energy change curve - distributed force

4.3 Cable Force and Ballast Adjustment in Cable-Stayed Bridges

This section demonstrates that cable forces and ballast in discrete systems satisfy constrained orthogonality and presents a generalized unit load method. When variable loads are free, displacements can be directly set to zero. When two forces between nodes are equal and opposite, orthogonality requires equal node displacements, allowing the large stiffness method (forcing zero displacement between nodes). For more complex variable load constraints, the generalized unit load method is proposed, illustrated using cable-stayed bridge ballast.

For analysis, a simplified cable-stayed bridge is considered with two towers, main span $L_1 + L_2 + L_1$, tower heights $H_{11}$ above and $H-2$ below deck, girder moment of inertia $EI_b$, tower moment of inertia $EI_t$, and 3 pairs of cables per tower. The elevation layout is shown below.

First, optimal cable forces are analyzed. Since cables are not parallel to displacement directions, the orthogonality condition is:
$$
T_{si}u_{ti} + T_{xi}w_{bi} = T_{si}u_{ti}\cos\theta_i + T_{xi}w_{bi}\sin\theta_i = 0
$$

Neglecting cable self-weight ($T_{si} = T_{xi}$) simplifies to:
$$
u_{ti}\cos\theta_i + w_{bi}\sin\theta_i = 0
$$

For side-span ballast (typically needed to address negative support reactions and main-span deflection), assume ballast loads $F_1, F_2, F_3$ satisfying:
$$
F_1 = 2F_2
$$

Let $F_2 = F$, then $F_1 = 2F$, reducing ballast loads to two independent variables $F, F_3$. Taking generalized unit loads ($F=1, F_3=1$) gives ballast values $2, 1, 1$. Since $F, F_3$ are independent, they can be considered separately. For $F_3$, directly set $w_3 = 0$. For $F$, orthogonality gives:
$$
2Fw_1 + Fw_2 = 0 \Rightarrow 2w_1 + w_2 = 0
$$

To find $F$, apply the ballast to displacements from original loads to get virtual work $V_0 = 2Fw_{10} + Fw_{20}$, then apply ballast alone to get $V_1 = 2Fw_{11} + Fw_{21}$. Orthogonality requires:
$$
V_0 + FV_1 = 0 \Rightarrow F = -\frac{V_0}{V_1}
$$

This process constitutes the generalized unit load method.

Conclusions

This paper proposes a minimum energy principle with variable loads and verifies it through multiple examples. Main conclusions:

1. Innovative principle: A minimum energy principle with variable loads is creatively proposed: system energy is minimized when virtual work of variable loads equals zero, i.e., when variable loads are orthogonal to their displacements.

2. Fundamental equations: Basic equations for mechanical systems with variable loads are derived, including geometric, equilibrium, constitutive, and boundary conditions, plus the orthogonality condition as an additional control equation. Without this condition, they reduce to conventional equations for fixed-boundary systems.

3. Transformation: The proposed principle transforms variable load control into displacement control, eliminating the need for conventional optimization methods and enabling seamless integration with finite element analysis.

4. Computational methods: Specific methods for structural optimization with variable loads are proposed: forced displacement method for freely variable loads, large stiffness method for equal-and-opposite loads, and generalized unit load method for complex constraints.

5. Validation: Multiple simplified examples verify the theory and provide optimization methods and design guidance, demonstrating broad applicability and practical value for structural optimization.

This paper systematically presents a minimum energy principle with variable loads, derives orthogonality conditions, transforms variable load optimization into displacement boundary conditions, and generalizes energy principles and fundamental mechanical equations. It offers high theoretical and practical value for optimization design of bridges and other structural engineering applications, advancing structural optimization development.

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