Numerical Investigation of Diagonals in Miter Gates: Looking for the Optimum Prestressing

In the last three decades, miter gates (Fig. 1) have experienced a significant amount of distress, the most common being fatigue cracking caused at welded connections. Other problems, such as pintle cracking, buckling plates, out-of-plane distortion, fatigue failure of diagonals, and extensive crack propagation were the result of the deterioration of the design boundary conditions (Riveros and Arredondo 2011; Riveros et al. 2009; Mahmoud et al. 2014; Mahmoud and Riveros 2014). In recent years, there have been a number of failures of diagonals in miter gates (Fig. 2). The authors hypothesize that these diagonal failures are fatigue-induced, driven by the connection details at the diagonal ends, but some evidence also suggests that the current design guidance (Hoffmann 1944; Shermer 1951; Headquarters, U. S. Army Corps of Engineers (USACE) 1994, 2014) results in a much larger prestress in diagonals than may be required. This article presents a three dimensional (3D)-implicit numerical simulation to assess the diagonal prestressing levels, which will provide the beginning for a new design approach. The new techniques will allow the prediction of optimum prestressing levels without compromising the fatigue strength. These results will be compared with the 1944, 1951, 1957, 1994, and 2014 criteria used by agencies responsible for maintenance and operation of miter gates.

Lock gates serve different functions, depending on location and condition. The major use of lock gates is to form a damming surface across a lock chamber, but the gates may also be used as guard gates, to fill and empty a lock chamber and to allow ice and debris to pass. A navigation lock requires closure gates at both ends so that the water level in the lock chamber can be varied to coincide with the upper and lower approach channels. Many locks in the United States are equipped with double-leaf miter gates that are used for moderate- and high-lift locks, with a height of 6–24 m and a chamber width of 17–33.5 m (Fig. 3).

A miter gate is a very deep cantilever girder with a relatively short span. The skin plate is the web of this girder. If the ordinary formulas for the deflection of a cantilever under shearing and bending stresses are applied, the vertical deflection of the miter gate leaf will be found as only a few tenths of a millimeter. Because the skin plate imparts such an abundant vertical stiffness to the leaf, the stresses in the diagonals are a function of only the torsional forces acting upon the leaf. These forces produce a considerable torsional deflection when the gate is being operated (USAED 1960).

The diagonals must be prestressed to maintain adequate torsional rigidity, typically during operation. The main functions of the diagonals prestressing is to reduce the torsional deflection of the leaf, to minimize the shear stress on the pintle region, to maintain the gate within tolerances regarding the vertical displacement, and to reduce the geometrical torsion (USACE 1994).

The Hoffman (1944), Shermer (1951, 1957), and USACE (1994, 2014) criteria define the shape of the twisted leaf geometrically. Then, the work done by the loads is equated to the internal work realized by the structure. From this, the resistance offered by each diagonal to twisting of the leaf is calculated as a function of the torsional deflection of the structure. The deflection is positive when the top of the leaf moves upstream in relation to the bottom. With the positive deflection, those diagonals that decrease in length are considered positive diagonals. The deflection is negative when the top of the gate moves downstream in relation to the bottom. With the negative deflection, those diagonals that decrease in length are considered negative diagonals (Fig. 4).

Hoffman (1944), Shermer (1951, 1957), and USACE (1994, 2014) assumed that the diagonals carry the vertical shear in a panel similar to the lateral bracing on buildings. The diagonals on a miter gate transfer vertical shear to the supports at the quoin and miter ends (Riveros 1995) and can be seen in Fig. 4. The stresses due to the dead load are assumed to be resisted by the truss formed by the top and bottom horizontal girders, the vertical quoin and miter post, and the skin plate and diagonal system. They also assumed that diagonal tensile stresses are divided equally between the skin plate and the diagonals on the opposite side of the gate. Fig. 4 shows a typical miter gate leaf reflecting the 1944, 1951, 1957, 1994, and 2014 design provisions.

The process described by the 1944, 1951, 1957, 1994, and 2014 criteria to design the diagonals first defines the stiffness of the leaf in deforming the diagonal ($A^{\prime }$). This value is conservative because it is taken as the sum of the average cross-sectional areas of the two-end diaphragms and the top and bottom girders, which bound a panel; this value must be increased by 12.5% for the heavier, welded, and horizontally framed leaves. For this example, $A^{\prime }$ is $287{\mathrm{cm}}^2$. Secondly, the elasticity constant $Q_o$, which is a measure of the torsional stiffness of a leaf without diagonals, is determined with a constant obtained from a limited amount of experiments conducted in 1944. $Q_o$ is $\mathrm{5,609}\mathrm{k}\mathrm{N}·\mathrm{m}$ for the present example. Thirdly, the load torque area ($Tz$), which is the product of an applied load ($F$); the distance ($z$) from the pintle to the applied load; and the moment arm ($r$) of the applied load with respect to the center of moments (located at the operating strut elevation) (Figs. 4 and 5) are determined. Table 1 shows the load torque areas with the corresponding distances for this example. The loads considered are the dead load, operating load, hydrodynamic load, and the temporal load that represent differential heads, wind waves, ship waves, and propeller wash. Fourthly, the ratio of change in length of diagonal to the deflection of the leaf when diagonal offers no resistance ($R_o$) is obtained. $R_o$ is positive for positive diagonals and negative for negative diagonals. For the present example, $R_o=\pm 0.1238$. The summation of the maximum positive and negative load torque areas ($\mathrm{\Sigma }{T}_z$) will provide the total area required in the set of negative and positive diagonals ($A_p$ and $A_n$) for the predefined load combinations. For this example, $A_p=483{\mathrm{cm}}^2$ and $A_n=419{\mathrm{cm}}^2$. The ratios of the actual change in length of the diagonal to the prestress deflection of the leaf ($R$) are $R_p=0.00461$ and $R_n=-0.0503$.

The hydrodynamic load $(H_d)$ is the inertial resistance to water while a leaf is operated. From tests performed to determine operating machinery design loads, the maximum value of the hydrodynamic load was found to be equivalent to a resistance of 1.44 kPa on the submerged portion of the gate leaf.

Once $A^{\prime }$, $Q_0$, $T_z$, $R_0$, $A_p$, $A_n$, $R_p$, $R_n$, $Q_n$, and $Q_p$ are calculated, the elasticity constant of the diagonal $Q$ is defined. The elasticity constant of the diagonal ($Q$) is a measure of the resistance in which a diagonal member offers to the torsional deflection of a leaf. $Q$ for this example is $Q_p=8.30\times {10}^3\mathrm{k}\mathrm{N}·\mathrm{m}$ and $Q_n=7.85\times {10}^3\mathrm{k}\mathrm{N}·\mathrm{m}$. Then, the maximum positive and maximum negative deflections of the leaf are calculated as ${\mathrm{\Delta }}_{\mathrm{opening}}=183\mathrm{mm}$ and ${\mathrm{\Delta }}_{\mathrm{closing}}=-183\mathrm{mm}$. The dead load will cause no torsional deflection because the gate leaf will be adjusted so that it has no vertical deformation ($D=0$) under dead load. If the gate leaf is to hang plumb under torsional action, the following equation is to be satisfied:where ${(T_z)}_D$ = dead load torque area; $Q$ = elasticity constant of a diagonal; and $D$ = prestress deflection for a diagonal.

The values of $D$ must be selected so that they follow the statement specified in Eq. (1). For a negative diagonal, let $D=-201\mathrm{mm}$, then, $QD=-1.57\times {10}^3\mathrm{k}\mathrm{N}·{\mathrm{m}}^2$. The dead load torque area is $-2.54\times {10}^3\mathrm{k}\mathrm{N}·{\mathrm{m}}^2$. In order to satisfy Eq. (1), the quantity $QD$ for the positive diagonal $D$ must equal 220 mm to guarantee that the leaf will not have vertical deformation under dead load. Prestressed deflections and stress in diagonals follow Eq. (2). The required stresses in the diagonals during operation are described in Tables 2 and 3where $s$ = unit stress in diagonal; $R$ = ratio of actual change length of diagonal to deflection of leaf when diagonal offers no resistance; $E$ = modulus of elasticity; $D$ = prestress deflection for a diagonal; $\mathrm{\Delta }$ = total torsional deflection of the leaf measured at the miter end, by the movement of the top girder relative to the bottom girder; and $L$ = length of diagonal.

The design criteria only takes into account the top horizontal girder, bottom horizontal girder, the vertical quoin, and miter posts, the skin plate, and diagonal system contributing to the rotational stiffness of the gate require a higher prestressing. The proposed numerical approach will consider all of the miter gate elements contributing to the torsional stiffness of the gate.

A series of analyses were carried out using the 3D-finite-element model of a miter gate shown in Fig. 10. The model consisted of 3D shell elements with six degrees of freedom per node. There are two main numerical experiments that will be discussed. The first is gravity with positive and negative diagonals (Case 1). The second is gravity with positive and negative diagonals, hydrostatic resistance pressure, and operation of the gate (Case 2). For both experiments, different prestressing diagonals were evaluated.

Displacement and loads boundary conditions: During the filling and lowering of the chamber pool, the gate is restrained in the 1, 2, and 3 directions in the pintle and in the 1 and 2 directions in the gudgeon pin, quoin block, and miter block (Fig. 10). The dead load, diagonals prestressing, and hydrostatic pressure induced by the upper and lower pool define the load boundary conditions. For this analysis, the upper pool is steadily lowered until it reaches hydraulic equilibrium. During gate operation, the displacement boundary conditions are located in the pintle and gudgeon pin. The load boundary conditions for the gate operation are the self-weight of the gate, prestressing diagonals, and water resistance pressure of 1.4 kPa (USACE 1994).

The comparison of the existing criteria developed in 1944, 1951, 1957, 1994, and 2014 to the numerical experiments shows that the numerical investigation can be used to obtain the optimum prestressing for miter gate diagonals. The comprehensive miter gate diagonals prestress analysis demonstrated the importance of maintaining the adequate levels of prestress. Opening and closing the gate produced an additional lateral displacement. Relaxation will develop over time, and vertical and lateral displacement on the miter end will increase, causing misalignment and additional torsional stresses on the pintle region. The magnitude of the misalignments can be obtained graphically from the developed curves.

Optimum values of prestressing can easily be calculated using numerical simulations. Once the different diagonal prestressing models are obtained, a series of curves relating prestressing levels with relative displacements can be accomplished. Then, for a predefined tolerance, the optimum prestressing levels can be easily extracted.

When the gate was closing, it required 161 and 8.0 MPa of positive and negative prestressing. This displacement was relatively large compared to the displacements from the numerical investigation.

The 1944, 1951, 1957, 1994, and 2014 criteria limited its design procedure of the amount of elements that can contribute to stiffening of the gate. In comparing the 1944, 1951, 1957, 1994, and 2014 criteria to the numerical simulation, the authors found that there was no one set of prestressing combinations for the positive and negative diagonals.

Finally, the numerical solution provides a reduction of 14 and 17% for the positive and negative diagonal prestressing. This indicates that the current design guidance results in larger prestress in miter gate diagonals than is required. Therefore, if this new approach to obtain the optimum diagonal prestressing is adopted, miter gates’ diagonals and their connections should have a better performance.